ytt^" 


?o-o 


IN  THE  SAME  SERIES. 


ON  THE  STUDY  AND  DIFFICULTIES  OF  MATHE- 
MATICS. By  AUGUSTUS  DE  MORGAN.  Entirely  new  edi- 
tion, with  portrait  of  the  author,  index,  and  annotations, 
bibliographies  of  modern  works  on  algebra,  the  philosophy 
of  mathematics,  pan-geometry,  etc.  Pp.,  288.  Cloth,  81.25 
net  (55.). 

LECTURES  ON  ELEMENTARY  MATHEMATICS.  By 
JOSEPH  Louis  LAGRANGE.  Translated  from  the  French  by 
Thomas  J.  McCormack.  With  photogravure  portrait  of 
Lagrange,  notes,  biography,  marginal  analyses,  etc.  Only 
separate  edition  in  French  or  English.  Pages,  172.  Cloth, 
$1.00  net  (ss.). 

ELEMENTARY  ILLUSTRATIONS  OF  THE  DIFFEREN- 
TIAL AND  INTEGRAL  CALCULUS.  By  AUGUSTUS  DE 
MORGAN.  New  reprint  edition.  With  sub-headings,  and 
a  brief  bibliography  of  English,  French,  and  German  text- 
books of  the  Calculus.  Pp.,  144.  Price,  $1.00  net  (55.)- 

MATHEMATICAL  ESSAYS  AND  RECREATIONS.  By 
HERMANN  SCHUBERT,  Professor  of  Mathematics  in  the 
Johanneum,  Hamburg,  Germany.  Translated  from  the 
German  by  Thomas  J.  McCormack.  Containing  essays  on 
The  Notion  and  Definition  of  Number,  Monism  in  Arith- 
metic, The  Nature  of  Mathematical  Knowledge,  The 
Magic  Square,  The  Fourth  Dimension,  The  Squaring  of 
the  Circle.  Pages,  149.  Cuts,  37.  Price,  Cloth,  750  net 
(3S.  6d.). 

A  BRIEF  HISTORY  OF  ELEMENTARY  MATHEMATICS. 
By  DR.  KARL  FINK,  late  Professor  in  Tubingen.  Translated 
from  the  German  by  Prof.  Wooster  Woodruff  Beman  and 
Prof.  David  Eugene  Smith.  Pp.  333.  Price,  cloth,  $1.50 
net  (6s.).  

THE  OPEN  COURT  PUBLISHING  CO. 

324  DEARBORN  ST.,   CHICAGO. 


A  BRIEF 


HISTORY  OF  MATHEMATICS 


&NSLATION  OF 


DR.   KARL_FINK'S  GESCHICHTE  DER 
ELEMENTAR-MATHEMATIK 


WOOSTER  WOODRUFF  BEMAN 

PROFESSOR  OP  MATHEMATICS  IN  THE  UNIVERSITY  OF  Ml' 


DAVID  EUGENE  SMITH 

PRINCIPAL  OF  THE  STATE  NORMAL  SCHOOL  AT  BROCKPORT,  N. 


CHICAGO 

THE  OPEN  COURT  PUBLISHING  COMPANY 

LONDON  AGENTS 

KEGAN  PAUL,  TRENCH,  TRUBNER  &  Co.,  LTD. 
1900 


TRANSLATION  COPYRIGHTED 
BY 

THE  OPEN  COURT  PUBLISHING  Co. 
1900. 


UNIVERSITY  OF  CALIFORNIA 
SANTA  BARBARA  COLLEGE  LIBRARY 


TRANSLATORS'  PREFACE. 

'TVHE  translators  feel  that  no  apology  is  necessary  for  any  rea- 
•*•  sonable  effort  to  encourage  the  study  of  the  history  of  mathe- 
matics. The  clearer  view  of  the  science  thus  afforded  the  teacher, 
the  inspiration  to  improve  his  methods  of  presenting  it,  the  in- 
creased interest  in  the  class-work,  the  tendency  of  the  subject  to 
combat  stagnation  of  curricula,— these  are  a  few  of  the  reasons  for 
approving  the  present  renaissance  of  the  study. 

This  phase  of  scientific  history  which  Montucla  brought  into 
such  repute — it  must  be  confessed  rather  by  his  literary  style  than 
by  his  exactness — and  which  writers  like  De  Morgan  in  England, 
Chasles  in  France,  Quetelet  in  Belgium,  Hankel  and  Baltzer  in 
Germany,  and  Boncompagni  in  Italy  encouraged  as  the  century 
wore  on,  is  seeing  a  great  revival  in  our  day.  This  new  movement 
is  headed  by  such  scholars  as  Gunther,  Enestrom,  Loria,  Paul 
Tannery,  and  Zeuthen,  but  especially  by  Moritz  Cantor,  whose 
Vorlesungen  fiber  Geschichte  der  Mathematik  must  long  remain 
the  world's  standard. 

In  any  movement  of  this  kind  compendia  are  always  necessary 
for  those  who  lack  either  the  time  or  the  linguistic  power  to  read 
the  leading  treatises.  Several  such  works  have  recently  appeared 
in  various  languages.  But  the  most  systematic  attempt  in  this 
direction  is  the  work  here  translated.  The  writers  of  most  hand- 
books of  this  kind  feel  called  upon  to  collect  a  store  of  anecdotes, 
to  incorporate  tales  of  no  historic  value,  and  to  minimize  the  real 
history  of  the  science.  Fink,  on  the  other  hand,  omits  biography 
entirely,  referring  the  reader  to  a  brief  table  in  the  appendix  or  to 
the  encyclopedias.  He  systematically  considers  the  growth  of 


rv  HISTORY  OF  MATHEMATICS. 

arithmetic,  algebra,  geometry,  and  trigonometry,  carrying  the  his- 
toric development,  as  should  be  done,  somewhat  beyond  the  limits 
of  the  ordinary  course. 

At  the  best,  the  work  of  the  translator  is  a  rather  thankless 
task.  It  is  a  target  for  critics  of  style  and  for  critics  of  matter. 
For  the  style  of  the  German  work  the  translators  will  hardly  be 
held  responsible.  It  is  not  a  fluent  one,  leaning  too  much  to  the 
scientific  side  to  make  it  always  easy  reading.  Were  the  work 
less  scientific,  it  would  lend  itself  more  readily  to  a  better  English 
form,  but  the  translators  have  preferred  to  err  on  the  side  of  a 
rather  strict  adherence  to  the  original. 

As  to  the  matter,  it  has  seemed  unwise  to  make  any  consider- 
able changes.  The  attempt  has  been  made  to  correct  a  number  of 
unquestionable  errors,  occasional  references  have  been  added,  and 
•  the  biographical  notes  have  been  rewritten.  It  has  not  seemed 
advisable,  however,  to  insert  a  large  number  of  bibliographical 
notes.  Readers  who  are  interested  in  the  subject  will  naturally 
place  upon  their  shelves  the  works  of  De  Morgan,  Allman,  Gow, 
Ball,  Heath,  and  other  English  writers,  and,  as  far  as  may  be, 
works  in  other  languages.  The  leading  German  authorities  are 
mentioned  in  the  footnotes,  and  the  French  language  offers  little 
at  present  beyond  the  works  of  Chasles  and  Paul  Tannery. 

The  translators  desire  to  express  their  obligations  to  Professor 
Markley  for  valuable  assistance  in  the  translation. 

Inasmuch  as  the  original  title  of  the  work,  Geschichte  der 
Elementar-Mathematik,  is  misleading,  at  least  to  English  read- 
ers, the  work  going  considerably  beyond  the  limits  of  the  elements, 
it  has  been  thought  best  to  use  as  the  English  title,  A  Brief  His 
tory  of  Mathematics. 

W.  W.  BEMAN,  Ann  Arbor,  Mich 
D.  E.  SMITH,  Brockport.  N.  Y. 
March,  1900. 


PREFACE. 

TF  the  history  of  a  science  possesses  value  for  every  one  whom 
•*•  calling  or  inclination  brings  into  closer  relations  to  it, — if  the 
knowledge  of  this  history  is  imperative  for  all  who  have  influence 
in  the  further  development  of  scientific  principles  or  the  methods 
of  employing  them  to  advantage,  then  acquaintance  with  the  rise 
and  growth  of  a  branch  of  science  is  especially  important  to  the 
man  who  wishes  to  teach  the  elements  of  this  science  or  to  pene- 
trate as  a  student  into  its  higher  realms. 

The  following  history  of  elementary  mathematics  is  intended 
to  give  students  of  mathematics  an  historical  survey  of  the  ele- 
mentary parts  of  this  science  and  to  furnish  the  teacher  of  the  ele- 
ments opportunity,  with  little  expenditure  of  time,  to  review  con- 
nectedly points  for  the  most  part  long  familiar  to  him  and  to  utilise 
them  in  his  teaching  in  suitable  comments.  The  enlivening  in- 
fluence of  historical  remarks  upon  this  elementary  instruction  has 
never  been  disputed.  Indeed  there  are  text-books  for  the  elements 
of  mathematics  (among  the  more  recent  those  of  Baltzer  and  Schu- 
bert) which  devote  considerable  space  to  the  history  of  the  science 
in  the  way  of  special  notes.  It  is  certainly  desirable  that  instead 
of  scattered  historical  references  there  should  be  offered  a  con- 
nected presentation  of  the  history  of  elementary  mathematics,  not 
one  intended  for  the  use  of  scholars,  not  as  an  equivalent  for  the 
great  works  upon  the  history  of  mathematics,  but  only  as  a  first 
picture,  with  fundamental  tones  clearly  sustained,  of  the  principal 
results  of  the  investigation  of  mathematical  history. 

In  this  book  the  attempt  has  been  made  to  differentiate  the 
histories  of  the  separate  branches  of  mathematical  science.  There 


CONTENTS. 


PAGE 

Translators'  Preface iii 

Author's  Preface v 

General  Survey i 

I.  NUMBER-SYSTEMS  AND  NUMBER-SYMBOLS.       6 

II.  ARITHMETIC. 

A.  General  Survey 18 

B.  First  Period.     The  Arithmetic  of  the  Oldest  Nations  to 

the  Time  of  the  Arabs. 

1.  The  Arithmetic  of  Whole  Numbers 24 

2.  The  Arithmetic  of  Fractions 31 

3.  Applied  Arithmetic 34 

C.  Second  Period.     From  the  Eighth  to  the  Fourteenth  Cen- 

tury. 

1.  The  Arithmetic  of  Whole  Numbers 36 

2.  The  Arithmetic  of  Fractions 40 

3.  Applied  Arithmetic 41 

D.  Third  Period.    From  the  Fifteenth  to  the  Nineteenth  Cen- 

tury. 

1.  The  Arithmetic  of  Whole  Numbers 41 

2.  The  Arithmetic  of  Fractions 49 

3.  Applied  Arithmetic 51 

III.  ALGEBRA. 

A.  General  Survey 61 

B.  First  Period.     From  the  Earliest  Times  to  the  Arabs. 

i.   General  Arithmetic 63 

Egyptian  Symbolism  63.     Greek  Arithmetic  64;  Symbolism 
65;  Theory  of  Numbers  66;  Series  67;  the  Irrational  68;  Neg- 


HISTORY  OF  MATHEMATICS. 


PAGE 

ative  Numbers  70;  Archimedes's  Notation  for  Large  Numbers 
71.  Roman  Arithmetic  71.  Hindu  Arithmetic  71 ;  Symbolism 
72;  Negative  Numbers  72;  Involution  and  Evolution  73;  Per- 
mutations and  Combinations  74 ;  Series  74.  Chinese  Arith- 
metic 74.  Arab  Arithmetic  74 ;  "Algorism  "  75 ;  Radical  Signs 
76 ;  Theory  of  Numbers  76 ;  Series  76. 

2.  Algebra 77 

The  Egyptians  77.  The  Greeks;  Form  of  the  Equation  77; 
Equations  of  the  First  Degree  78 ;  Equations  of  the  Second 
Degree  {Application  of  Areas)  79;  Equations  of  the  Third  De- 
gree 81 ;  Indeterminate  Equations  (Cattle  Problem  of  Archi- 
medes; Methods  of  Solution  of  Diophantus)  83.  Hindu  Al- 
gebra 84.  Chinese  Algebra  87.  Arab  Algebra  88. 

C.  Second  Period.     To  the  Middle  of  the  Seventeenth  Cen- 

tury. 

1.  General  Arithmetic 95 

Symbolism  of  the  Italians  and  the  German  Cossists  95;  Irra- 
tional and  Negative  Numbers  99 ;  Imaginary  Quantities  101 ; 
Powers  102 ;  Series  103 ;  Stifel's  Duplication  of  the  Cube  104  ; 
Magic  Squares  105. 

2.  Algebra 107 

Representation  of  Equations  107;  Equations  of  the  First  and 
Second  Degrees  108 ;  Complete  Solution  of  Equations  of  the 
Third  and  Fourth  Degrees  by  the  Italians  in  ;  Work  of  the 
German  Cossists  113  ;  Beginnings  of  a  General  Theory  of  Al- 
gebraic Equations  115. 

D.  Third  Period.  From  the  Middle  of  the  Seventeenth  Cen- 

tury to  the  Present  Time. 

Symbolism  117;  Pascal's  Arithmetic  Triangle  118;  Irrational 
Numbers  119;  Complex  Numbers  123;  Grassmann's  Aut- 
deknvngslehre  127 ;  Quaternions  129;  Calculus  of  Logic  131; 
Continued  Fractions  131 ;  Theory  of  Numbers  133 ;  Tables  of 
Primes  141 ;  Symmetric  Functions  142;  Elimination  143  ;  The- 
ory of  Invariants  and  Covariants  145  ;  Theory  of  Probabilities 
148;  Method  of  Least  Squares  149;  Theory  of  Combinations 
150;  Infinite  Series  (Convergence  and  Divergence)  151 ;  Solu- 
tion of  Algebraic  Equations  155 ;  the  Cyclotomic  Equation 
160;  Investigations  of  Abel  and  Galois  163;  Theory  of  Substi- 
tutions 164;  the  Equation  of  the  Fifth  Degree  165;  Approxi- 
mation of  Real  Roots  166 ;  Determinants  107;  Differential  and 
Integral  Calculus  168;  Differential  Equations  174;  Calculus 
of  Variations  178  ;  Elliptic  Functions  180;  Abelian  Functions 
186;  More  Rigorous  Tendency  of  Analysis  189. 


CONTENTS. 


IV.  GEOMETRY. 

PAGE 

A.  General  Survey 190 

B.  First  Period.     Egyptians  and  Babylonians 192 

C.  Second  Period.     The  Greeks 193 

The  Geometry  oLTJiales  and  Pythagoras  194;  Application  of 
the  Quadratrix  to  the  Quadrature  of  the  Circle  and  the  Trisec- 
tion  of  an  Angle  196;  the  Elements  of  Euclid  198;  Archimedes 
and  his  Successors  199 ;  the  Theory  of  Conic  Sections  »oa; 
the  Duplication  of  the  Cube,  the  Trisection  of  an  Angle  and 
the  Quadrature  of  the  Circle  209;  Plane,  Solid,  and  Linear 
Loci  209;  Surfaces  of  the  Second  Order  212;  the  Stereo- 
graphic  Projection  of  Hipparchus  213. 

D.  Third  Period.     Romans,  Hindus,  Chinese,  Arabs  .     .     .     214 

E.  Fourth  Period.     From  Gerbert  to  Descartes 218 

Gerbert  and  Leonardo  218;  Widmann  and  Stifelaao;  Vieta 
and  Kepler  222 ;  Solution  of  Problems  with  but  One  Opening 
of  the  Compasses  225;  Methods  of  Projection  226. 

F.  Fifth  Period.     From  Descartes  to  the  Present    ....     228 

Descartes's  Analytic  Geometry  230;  Cavalieri's  Method  of  In- 
divisibles 234 ;  Pascal's  Geometric  Works  237;  Newton's  In- 
vestigations 239;  Cramer's  Paradox  240;  Pascal's  Limacon 
and  other  Curves  241 ;  Analytic  Geometry  of  Three  Dimen- 
sions 242;  Minor  Investigations  243;  Introduction  of  Projec- 
tive  Geometry  246 ;  Mobius'  s  Barycentrischer  Calciil  250 ;  Bel- 
lavitis's  Equipollences  250;  Pliicker's  Investigations  251; 
Steiner's  Developments  256;  Malfatti's  Problem  256;  Von 
Staudt's  Geometrie  der  Lage  258  ;  Descriptive  Geometry  259  ; 
Form-theory  and  Deficiency  of  an  Algebraic  Curve  261 ; 
Gauche  Curves  263  ;  Enumerative  Geometry  264  ;  Conformal 
Representation  266 ;  Differential  Geometry  (Theory  of  Curva- 
ture of  Surfaces)  267;  Non-Euclidean  Geometry  270;  Pseudo- 
Spheres  273  ;  Geometry  of  «  Dimensions  275 ;  Geometria  and 
Analysis  Situs  275;  Contact-transformations  276;  Geometric 
Theory  of  Probability  276;  Geometric  Models  277;  the  Math- 
ematics of  To-day  279. 


V.  TRIGONOMETRY. 

A.  General  Survey 281 

B.  First  Period.  From  the  Most  Ancient  Times  to  the  Arabs     282 

The  Egyptians  282.    The  Greeks  282.    The  Hindus  284     The 
Arabs  285. 


xii  HISTORY  OF  MATHEMATICS. 

PAGE 

C.  Second  Period.     From  the  Middle  Ages  to  the  Middle  of 

the  Seventeenth  Century 287 

Vieta  and  Regiomontanus  387;   Trigonometric  Tables  289; 
Logarithms  290. 

D.  Third  Period.    From  the  Middle  of  the  Seventeenth  Cen- 

tury to  the  Present 294 

Biographical  Notes 297 

Index 323 


GENERAL  SURVEY. 

/rTAHE  beginnings  of  the  development  of  mathemat- 
-*-  ical  truths  date  back  to  the  earliest  civilizations 
of  which  any  literary  remains  have  come  down  to  us, 
namely  the  Egyptian  and  the  Babylonian.  On  the 
one  hand,  brought  about  by  the  demands  of  practical 
life,  on  the  other  springing  from  the  real  scientific 
spirit  of  separate  groups  of  men,  especially  of  the 
priestly  caste,  arithmetic  and  geometric  notions  came 
into  being.  Rarely,  however,  was  this  knowledge 
transmitted  through  writing,  so  that  of  the  Babylo- 
nian civilization  we  possess  only  a  few  traces.  From 
the  ancient  Egyptian,  however,  we  have  at  least  one 
manual,  that  of  Ahmes,  which  in  all  probability  ap- 
peared nearly  two  thousand  years  before  Christ. 

The  real  development  of  mathematical  knowledge, 
obviously  stimulated  by  Egyptian  and  Babylonian  in- 
fluences, begins  in  Greece.  This  development  shows 
itself  predominantly  in  the  realm  of  geometry,  and 
enters  upon  its  first  classic  period,  a  period  of  no 
great  duration,  during  the  era  of  Euclid,  Archimedes, 
Eratosthenes,  and  Apollonius.  Subsequently  it  in- 
clines more  toward  the  arithmetic  side  ;  but  it  soon 
becomes  so  completely  engulfed  by  the  heavy  waves 


2  HISTORY  OF  MATHEMATICS. 

of  stormy  periods  that  only  after  long  centuries  and 
in  a  foreign  soil,  out  of  Greek  works  which  had  es- 
caped the  general  destruction,  could  a  seed,  new  and 
full  of  promise,  take  root. 

One  would  naturally  expect  to  find  the  Romans 
entering  with  eagerness  upon  the  rich  intellectual 
inheritance  which  came  to  them  from  the  conquered 
Greeks,  and  to  find  their  sons,  who  so  willingly  re- 
sorted to  Hellenic  masters,  showing  an  enthusiasm 
for  Greek  mathematics.  Of  this,  however,  we  have 
scarcely  any  evidence.  The  Romans  understood  very 
well  the  practical  value  to  the  statesman  of  Greek 
geometry  and  surveying — a  thing  which  shows  itself 
also  in  the  later  Greek  schools — but  no  real  mathe- 
matical advance  is  to  be  found  anywhere  in  Roman 
history.  Indeed,  the  Romans  often  had  so  mistaken 
an  idea  of  Greek  learning  that  not  infrequently  they 
handed  it  down  to  later  generations  in  a  form  entirely 
distorted. 

More  important  for  the  further  development  of 
mathematics  are  the  relations  of  the  Greek  teachings 
to  the  investigations  of  the  Hindus  and  the  Arabs. 
The  Hindus  distinguished  themselves  by  a  pronounced 
talent  for  numerical  calculation.  What  especially  dis- 
tinguishes them  is  their  susceptibility  to  the  influence 
of  Western  science,  the  Babylonian  and  especially 
the  Greek,  so  that  they  incorporated  into  their  own 
system  what  they  received  from  outside  sources  and 
then  worked  out  independent  results. 


GENERAL  SURVEY.  3 

The  Arabs,  however,  in  general  do  not  show  this 
same  independence  of  apprehension  and  of  judgment. 
Their  chief  merit,  none  the  less  a  real  one  however, 
lies  in  the  untiring  industry  which  they  showed  in 
translating  into  their  own  language  the  literary  treas- 
ures of  the  Hindus,  Persians  and  Greeks.  The  courts 
of  the  Mohammedan  princes  from  the  ninth  to  the 
thirteenth  centuries  were  the  seats  of  a  remarkable 
scientific  activity,  and  to  this  circumstance  alone  do 
we  owe  it  that  after  a  period  of  long  and  dense  dark- 
ness Western  Europe  was  in  a  comparatively  short 
time  opened  up  to  the  mathematical  sciences. 

The  learning  of  the  cloisters  in  the  earlier  part 
of  the  Middle  Ages  was  not  by  nature  adapted  to 
enter  seriously  into  matters  mathematical  or  to  search 
for  trustworthy  sources  of  such  knowledge.  It  was 
the  Italian  merchants  whose  practical  turn  and  easy 
adaptability  first  found,  in  their  commercial  relations 
with  Mohammedan  West  Africa  and  Southern  Spain, 
abundant  use  for  the  common  calculations  of  arith- 
metic. Nor  was  it  long  after  that  there  developed 
among  them  a  real  spirit  of  discovery,  and  the  first 
great  triumph  of  the  newly  revived  science  was  the 
solution  of  the  cubic  equation  by  Tartaglia.  It  should 
be  said,  however,  that  the  later  cloister  cult  labored 
zealously  to  extend  the  Western  Arab  learning  by 
means  of  translations  into  the  Latin. 

In  the  fifteenth  century,  in  the  persons  of  Peur- 
bach  and  Regiomontanus,  Germany  first  took  position 


4-i  HISTORY  OF  MATHEMATICS. 

in  the  great  rivalry  for  the  advancement  of  mathemat- 
ics. From  that  time  until  the  middle  of  the  seven- 
teenth century  the  German  mathematicians  were 
chiefly  calculators,  that  is  teachers  in  the  reckoning 
schools  (Rechenschuleri).  Others,  however,  were  alge- 
braists, and  the  fact  is  deserving  of  emphasis  that 
there  were  intellects  striving  to  reach  still  loftier 
heights.  Among  them  Kepler  stands  forth  pre-emi- 
nent, but  with  him  are  associated  Stifel,  Rudolff,  and 
Biirgi.  Certain  is  it  that  at  this  time  and  on  Ger- 
man soil  elementary  arithmetic  and  common  algebra, 
vitally  influenced  by  the  Italian  school,  attained  a 
standing  very  conducive  to  subsequent  progress. 

The  modern  period  in  the  history  of  mathematics 
begins  about  the  middle  of  the  seventeenth  century. 
Descartes  projects  the  foundation  theory  of  the  ana- 
lytic geometry.  Leibnitz  and  Newton  appear  as  the 
discoverers  of  the  differential  calculus.  The  time  has 
now  come  when  geometry,  a  science  only  rarely,  and 
even  then  but  imperfectly,  appreciated  after  its  ban- 
ishment from  Greece,  enters  along  with  analysis  upon 
a  period  of  prosperous  advance,  and  takes  full  advan- 
tage of  this  latter  sister  science  in  attaining  its  results. 
Thus  there  were  periods  in  which  geometry  was  able 
through  its  brilliant  discoveries  to  cast  analysis,  tem- 
porarily at  least,  into  the  shade. 

The  unprecedented  activity  of  the  great  Gauss 
divides  the  modern  period  into  two  parts :  before 
Gauss — the  establishment  of  the  methods  of  the  dif- 


GENERAL  SURVEY.  5 

ferential  and  integral  calculus  and  of  analytic  geom- 
etry as  well  as  more  restricted  preparations  for  later 
advance  ;  with  Gauss  and  after  him — the  magnificent 
development  of  modern  mathematics  with  its  special 
regions  of  grandeur  and  depth  previously  undreamed 
of.  The  mathematicians  of  the  nineteenth  century 
are  devoting  themselves  to  the  theory  of  numbers, 
modern  algebra,  the  theory  of  functions  and  projec- 
tive  geometry,  and  in  obedience  to  the  impulse  of 
human  knowledge  are  endeavoring  to  carry  their  light 
into  remote  realms  which  till  now  have  remained  in 
darkness. 


I.  NUMBER-SYSTEMS  AND  NUMBER- 
SYMBOLS. 

AN  inexhaustible  profusion  of  external  influences 
**•  upon  the  human  mind  has  found  its  legitimate 
expression  in  the  formation  of  speech  and  writing 
in  numbers  and  number-symbols.  It  is  true  that  a 
counting  of  a  certain  kind  is  found  among  peoples  of 
a  low  grade  of  civilization  and  even  among  the  lower 
animals.  "Even  ducks  can  count  their  young."*  But 
where  the  nature  and  the  condition  of  the  objects 
have  been  of  no  consequence  in  the  formation  of  the 
number  itself,  there  human  counting  has  first  begun. 
The  oldest  counting  was  even  in  its  origin  a  pro- 
cess of  reckoning,  an  adjoining,  possibly  also  in  special 
elementary  cases  a  multiplication,  performed  upon 
the  objects  counted  or  upon  other  objects  easily  em- 
ployed, such  as  pebbles,  shells,  fingers.  Hence  arose 
number-names.  The  most  common  of  these  undoubt- 
edly belong  to  the  primitive  domain  of  language  ;  with 
the  advancing  development  of  language  their  aggre- 
gate was  gradually  enlarged,  the  legitimate  combina- 


*Hankel,  Zur  Geschichte  der  Mathentatik  fm  Altertunt  und  Mittelalter 
1874,  p.  7.  Hereafter  referred  to  as  Hankel.  Tyler's  Primitive  Culture  alsc 
has  a  valuable  chapter  upon  counting. 


NUMBER-SYSTEMS  AND  NUMBER-SYMBOLS.  7 

tion  of  single  terms  permitting  and  favoring  the  crea- 
tion of  new  numbers.  Hence  arose  number-systems. 

The  explanation  of  the  fact  that  10  is  almost  every- 
where found  as  the  base  of  the  system  of  counting  is 
seen  in  the  common  use  of  the  fingers  in  elementary 
calculations.  In  all  ancient  civilizations  finger-reckon- 
ing was  known  and  even  to-day  it  is  carried  on  to  a 
remarkable  extent  among  many  savage  peoples.  Cer- 
tain South  African  races  use  three  persons  for  num- 
bers which  run  above  100,  the  first  counting  the  units 
on  his  fingers,  the  second  the  tens,  and  the  third  the 
hundreds.  They  always  begin  with  the  little  finger  of 
the  left  hand  and  count  to  the  little  finger  of  the  right. 
The  first  counts  continuously,  the  others  raising  a 
finger  every  time  a  ten  or  a  hundred  is  reached.* 

Some  languages  contain  words  belonging  funda- 
mentally to  the  scale  of  5  or  20  without  these  systems 
having  been  completely  elaborated  ;  only  in  certain 
places  do  they  burst  the  bounds  of  the  decimal  sys- 
tem. In  other  cases,  answering  to  special  needs,  12 
and  60  appear  as  bases.  The  New  Zealanders  have 
a  scale  of  11,  their  language  possessing  words  for  the 
first  few  powers  of  11,  and  consequently  12  is  repre- 
sented as  11  and  1,  13  as  11  and  2,  22  as  two  ll's, 
and  so  on.f 


*  Cantor,  M.,  Vorlcsungen  uber  Geschichte  der  Mathematik.  Vol.  I,  1880; 
2nd  ed.,  1894,  p.  6.  Hereafter  referred  to  as  Cantor.  Conant,  L.  L.,  The  Num- 
ber Concept,  N.Y.  1896.  Gow,  J.,  History  of  Greek  Geometry,  Cambridge,  1884, 
Chap.  I. 

t  Cantor,  I.,  p.  10. 


8  HISTORY  OF  MATHEMATICS. 

In  the  verbal  formation  of  a  number-system  addi- 
tion and  multiplication  stand  out  prominently  as  defin- 
itive operations  for  the  composition  of  numbers  ;  very 
rarely  does  subtraction  come  into  use  and  still  more 
rarely  division.  For  example,  18  is  called  in  Latin 
10 +  8  (decem  et  octo),  in  Greek  8  +  10  (oKTw-xai-ScKa), 
in  French  10  8  (dix-huif),  in  German  8  10  (acht-zehti), 
in  Latin  also  20  —  2  (duo-de-viginti*),  in  Lower  Breton 
3-6  (tri-omc'h'),  in  Welsh  2-9  (dew-naw),  in  Aztec 
15  -\-  3  (caxtulli-om-ey},  while  50  is  called  in  the  Basque 
half-hundred,  in  Danish  two-and-a-half  times  twenty.* 
In  spite  of  the  greatest  diversity  of  forms,  the  written 
representation  of  numbers,  when  not  confined  to  the 
mere  rudiments,  shows  a  general  law  according  to 
which  the  higher  order  precedes  the  lower  in  the  di- 
rection of  the  writing."}"  Thus  in  a  four-figure  number 
the  thousands  are  written  by  the  Phoenicians  at  the 
right,  by  the  Chinese  above,  the  former  writing  from 
right  to  left,  the  latter  from  above  downward.  A 
striking  exception  to  this  law  is  seen  in  the  sub 
tractive  principle  of  the  Romans  in  IV,  IX,  XL, 
etc.,  where  the  smaller  number  is  written  before  the 
larger. 

Among  the  Egyptians  we  have  numbers  running 
from  right  to  left  in  the  hieratic  writing,  with  varying 
direction  in  the  hieroglyphics.  In  the  latter  the  num- 
bers were  either  written  out  in  words  or  represented 
by  symbols  for  each  unit,  repeated  as  often  as  neces- 

*  Hankel.  p.  22.  tHankel,  p.  32. 


NUMBER-SYSTEMS  AND  NUMBER-SYMBOLS.  9 

sary.  In  one  of  the  tombs  near  the  pyramids  of  Gizeh 
have  been  found  hieroglyphic  numerals  in  which  1  is 
represented  by  a  vertical  line,  10  by  a  kind  of  horse- 
shoe, 100  by  a  short  spiral,  10  000  by  a  pointing  finger, 
100  000  by  a  frog,  1  000  000  by  a  man  in  the  attitude 
of  astonishment.  In  the  hieratic  symbols  the  figure 
for  the  unit  of  higher  order  stands  to  the  right  of  the 
one  of  lower  order  in  accordance  with  the  law  of  se- 
quence already  mentioned.  The  repetition  of  sym- 
bols for  a  unit  of  any  particular  order  does  not  obtain, 
because  there  are  special  characters  for  all  nine  units, 
all  the  tens,  all  the  hundreds,  and  all  the  thousands.* 
We  give  below  a  few  characteristic  specimens  of  the 
hieratic  symbols : 

I      II      III      -      1      AAV- 

13  3  4  5  10          20  80          40 

The  Babylonian  cuneiform  inscriptionsf  proceed 
from  left  to  right,  which  must  be  looked  upon  as  ex- 
ceptional in  a  Semitic  language.  In  accordance  with 
the  law  of  sequence  the  units  of  higher  order  stand  on 
the  left  of  those  of  lower  order.  The  symbols  used 
in  writing  are  chiefly  the  horizontal  wedge  >-,  the  ver- 
tical wedge  Y,  and  the  combination  of  the  two  at  an 
angle  .4.  The  symbols  were  written  beside  one  another, 
or,  for  ease  of  reading  and  to  save  space,  over  one 
another.  The  symbols  for  1,  4,  10,  100,  14,  400,  re- 
spectively, are  as  follows  : 

*  Cantor,  I.,  pp.  43,  44.  t  Cantor,  I.,  pp.  77,  78. 


HISTORY  OF  MATHEMATICS. 
% 


.vvv. 


1  4  10  100  14  400 

For  numbers  exceeding  100  there  was  also,  besides 
the  mere  juxtaposition,  a  multiplicative  principle ; 
the  symbol  representing  the  number  of  hundreds  was 
placed  at  the  left  of  the  symbol  for  hundreds  as  in  the 
case  of  400  already  shown.  The  Babylonians  probably 
had  no  symbol  for  zero.*  The  sexagesimal  system 
(i.  e.,  with  the  base  60),  which  played  such  a  part  in 
the  writings  of  the  Babylonian  scholars  (astronomers 
and  mathematicians),  will  be  mentioned  later. 

The  Phoenicians,  whose  twenty-two  letters  were 
derived  from  the  hieratic  characters  of  the  Egyptians, 
either  wrote  the  numbers  out  in  words  or  used  special 
numerical  symbols — for  the  units  vertical  marks,  for 
the  tens  horizontal,  f  Somewhat  later  the  Syrians  used 
the  twenty-two  letters  of  their  alphabet  to  represent 
the  numbers  1,  2,  .  .  9,  10,  20,  ...  90,  100,  ...  400  ; 
500  was  400 -f  100,  etc.  The  thousands  were  repre 
sented  by  the  symbols  for  units  with  a  subscript 
comma  at  the  right.  J  The  Hebrew  notation  follows 
the  same  plan. 

The  oldest  Greek  numerals  (aside  from  the  written 
words)  were,  in  general,  the  initial  letters  of  the  funda 
mental  numbers.  I  for  1,  n  for  5  (irore),  A  for  10 
(Se'Ka),§  and  these  were  repeated  as  often  as  necessary. 

•Cantor,  I.,  p.  84.         t Cantor,  I.,  p.  113.          t  Cantor.  I.,  pp.  113-114. 
{Cantor,  I.,  p.  no. 


NUMBER-SYSTEMS  AND  NUMBER-SYMBOLS.  I  I 

These  numerals  are  described  by  the  Byzantine  gram- 
marian Herodianus  (A.  D.  200)  and  hence  are  spoken 
of  as  Herodianic  numbers.  Shortly  after  500  B.  C. 
two  new  systems  appeared.  One  used  the  24  letters 
of  the  Ionic  alphabet  in  their  natural  order  for  the 
numbers  from  1  to  24.  The  other  arranged  these 
letters  apparently  at  random  but  actually  in  an  order 
fixed  arbitrarily;  thus,  o  =  l,  ft  =  2, .  .  . .  ,  t  =  10,  K  = 
20,  .  .  .  .  ,  p  =  100,  o-^200,  etc.  Here  too  there  is 
no  special  symbol  for  the  zero. 

The  Roman  numerals*  were  probably  inherited 
from  the  Etruscans.  The  noteworthy  peculiarities 
are  the  lack  of  the  zero,  the  subtractive  principle 
whereby  the  value  of  a  symbol  was  diminished  by 
placing  before  it  one  of  lower  order  (IV  =  4,  IX  =  9, 
XL  =  40,  XC  =  90),  even  in  cases  where  the  language 
itself  did  not  signify  such  a  subtraction ;  and  finally 
the  multiplicative  effect  of  a  bar  over  the  numerals 
(x3E?==30  000,  0  =  100000).  Also  for  certain  frac- 
tions there  were  special  symbols  and  names.  Accord- 
ing to  Mommsen  the  Roman  number-symbols  I,  V, 
X  represent  the  finger,  the  hand,  and  the  double 
hand.  Zangemeister  proceeds  from  the  standpoint 
that  decem  is  related  to  decussare  which  means  a 
perpendicular  or  oblique  crossing,  and  argues  that 
every  straight  or  curved  line  drawn  across  the  symbol 
of  a  number  in  the  decimal  system  multiplies  that  • 
number  by  ten.  In  fact,  there  are  on  monuments 

*  Cantor,  I.,  p.  486. 


12  HISTORY  OF  MATHEMATICS. 

representations  of  1,  10,  and  1000,  as  well  as  of  5  and 
500,  to  prove  his  assertion.* 

Of  especial  interest  in  elementary  arithmetic  is  the 
number-system  of  the  Hindus,  because  it  is  to  these 
Aryans  that  we  undoubtedly  owe  the  valuable  position- 
system  now  in  use.  Their  oldest  symbols  for  1  to  9 
were  merely  abridged  number-words,  and  the  use  of 
letters  as  figures  is  said  to  have  been  prevalent  from 
the  second  century  A.  D.f  The  zero  is  of  later  origin  ; 
its  introduction  is  not  proven  with  certainty  till  after 
400  A.  D.  The  writing  of  numbers  was  carried  on, 
chiefly  according  to  the  position-system,  in  various 
ways.  One  plan,  which  Aryabhatta  records,  repre- 
sented the  numbers  from  1  to  25  by  the  twenty-five 
consonants  of  the  Sanskrit  alphabet,  and  the  succeed- 
ing tens  (30,  40  ....  100)  by  the  semi-vowels  and 
sibilants.  A  series  of  vowels  and  diphthongs  formed 
multipliers  consisting  of  powers  of  ten,  ga  meaning 
3,  gi  300,  gu  30  000,  gau  3-1016.J  In  this  there  is  no 
application  of  the  position-system,  although  it  ap- 
pears in  two  other  methods  of  writing  numbers  in 
use  among  the  arithmeticians  of  Southern  India. 
Both  of  these  plans  are  distinguished  by  the  fact  that 

*SitnungsbericJite  der  Berliner  Akademie  vain  10,  November  1887 .  Words- 
worth, in  his  Fragments  and  Specimens  of  Early  Latin,  1874,  derives  C  for 
centum,  M  for  mille,  and  L  for  quinquaginta  from  three  letters  of  the  Chal- 
cidian  alphabet,  corresponding  to  9,  #,  and  x-  He  says:  "The  origin  of  this 
notation  is,  I  believe,  quite  uncertain,  or  rather  purely  arbitrary,  though,  of 
course,  we  observe  that  the  initials  of  mille  and  centum  determined  the  final 
shape  taken  by  the  signs,  which  at  first  were  very  different  in  form." 

tSee  Encyclopedia  Britannica,  under  "Numerals  " 

J  Cantor,  I.,  p.  566. 


NUMBER-SYSTEMS  AND  NUMBER-SYMBOLS.  13 

the  same  number  can  be  made  up  in  various  ways. 
Rules  of  calculation  were  clothed  in  simple  verse  easy 
to  hold  in  mind  and  to  recall.  For  the  Hindu  mathe- 
maticians this  was  all  the  more  important  since  they 
sought  to  avoid  written  calculation  as  far  as  possible. 
One  method  of  representation  consisted  in  allowing 
the  alphabet,  in  groups  of  9  symbols,  to  denote  the 
numbers  from  1  to  9  repeatedly,  while  certain  vowels 
represented  the  zeros.  If  in  the  English  alphabet  ac- 
cording to  this  method  we  were  to  denote  the  num- 
bers from  1  to  9  by  the  consonants  b,  c,  .  .  .  z  so  that 
after  two  countings  one  finally  has  z  =  2,  and  were  to 
denote  zero  by  every  vowel  or  combination  of  vowels, 
the  number  60502  might  be  indicated  by  siren  or  heron, 
and  might  be  introduced  by  some  other  words  in  the 
text.  A  second  method  employed  type-words  and 
combined  them  according  to  the  law  of  position. 
Thus  abdhi  (one  of  the  4  seas)  =  4,  surya  (the  sun 
with  its  12  houses)=12,  apvin  (the  two  sons  of  the 
sun)=2.  The  combination  abdhisurya$vina$  denoted 
the  number  2124.* 

Peculiar  to  the  Sanskrit  number-language  are  spe- 
cial words  for  the  multiplication  of  very  large  num- 
bers. Arbuda  signifies  100  millions,  padma  10000 
millions;  from  these  are  derived  maharbuda  =  1000 
millions,  mahapadma  =  \W  000  millions.  Specially- 
formed  words  for  large  numbers  run  up  to  1017  and 
even  further.  This  extraordinary  extension  of  the 

*  Cantor,  I.,  p.  567. 


14  HISTORY  OF  MATHEMATICS. 

decimal  system  in  Sanskrit  resembles  a  number-game, 
a  mania  to  grasp  the  infinitely  great.  Of  this  endeavor 
to  bring  the  infinite  into  the  realm  of  number-percep- 
tion and  representation,  traces  are  found  also  among 
the  Babylonians  and  Greeks.  This  appearance  may 
find  its  explanation  in  mystic-religious  conceptions  or 
philosophic  speculations. 

The  ancient  Chinese  number-symbols  are  confined 
to  a  comparatively  few  fundamental  elements  arranged 
in  a  perfectly  developed  decimal  system.  Here  the 
combination  takes  place  sometimes  by  multiplica- 
tion, sometimes  by  addition.  Thus  san  =  3,  c/ie  =  lQ; 
che  san  denotes  13,  but  san  che  30.*  Later,  as  a  result 
of  foreign  influence,  there  arose  two  new  kinds  of  no- 
tation whose  figures  show  some  resemblance  to  the 
ancient  Chinese  symbols.  Numbers  formed  from 
them  were  not  written  from  above  downward  but 
after  the  Hindu  fashion  from  left  to  right  beginning 
with  the  highest  order.  The  one  kind  comprising  the 
merchants'  figures  is  never  printed  but  is  found  only 
in  writings  of  a  business  character.  Ordinarily  the 
ordinal  and  cardinal  numbers  are  arranged  in  two 
lines  one  above  another,  with  zeros  when  necessary, 
in  the  form  of  small  circles.  In  this  notation 
||=2,X  =  4,  j_  =  6, -|.  =  10,  77  =  10000,  O  =  «, 

"  X 

and  hence    ft  O  O  -f-j_  =20046. 

*  Cantor.  I.,  p.  630. 


NUMBER-SYSTEMS  AND  NUMBER-SYMBOLS.  15 

Among  the  Arabs,  those  skilful  transmitters  of 
Oriental  and  Greek  arithmetic  to  the  nations  of  the 
West,  the  custom  of  writing  out  number-words  con- 
tinued till  the  beginning  of  the  eleventh  century. 
Yet  at  a  comparatively  early  period  they  had  already 
formed  abbreviations  of  the  number- words,  the  Divani 
figures.  In  the  eighth  century  the  Arabs  became  ac- 
quainted with  the  Hindu  number-system  and  its  fig- 
ures, including  zero.  From  these  figures  there  arose 
among  the  Western  Arabs,  who  in  their  whole  litera- 
ture presented  a  decided  contrast  to  their  Eastern  re- 
latives, the  Gubar  numerals  (dust-numerals)  as  vari- 
ants. These  Gubar  numerals,  almost  entirely  forgotten 
to-day  among  the  Arabs  themselves,  are  the  ancestors 
of  our  modern  numerals,*  which  are  immediately  de- 
rived from  the  apices  of  the  early  Middle  Ages.  These 
primitive  Western  forms  used  in  the  abacus-calcula- 
tions are  found  in  the  West  European  MSS.  of  the 
eleventh  and  twelfth  centuries  and  owe  much  of  their 
prominence  to  Gerbert,  afterwards  Pope  Sylvester  II. 
(consecrated  999  A.  D.). 

The  arithmetic  of  the  Western  nations,  cultivated 
to  a  considerable  extent  in  the  cloister-schools  from 
the  ninth  century  on,  employed  besides  the  abacus  the 
Roman  numerals,  and  consequently  made  no  use  of  a 
symbol  for  zero.  In  Germany  up  to  the  year  1500  the 
Roman  symbols  were  called  German  numerals  in  dis- 
tinction from  the  symbols — then  seldom  employed — 

"Hankel,  p.255- 


1 6  HISTORY  OF  MATHEMATICS. 

of  Arab-Hindu  origin,  which  included  a  zero  (Arabic 
as-sifr,  Sanskrit  sunya,  the  void).  The  latter  were 
called  ciphers  (Zifferri).  From  the  fifteenth  century  on 
these  Arab-Hindu  numerals  appear  more  frequently  in 
Germany  on  monuments  and  in  churches,  but  at  that 
time  they  had  not  become  common  property.*  The 
oldest  monument  with  Arabic  figures  (in  Katharein 
near  Troppau)  is  said  to  date  from  1007.  Monuments 
of  this  kind  are  found  in  Pforzheim  (1371),  and  in  Ulm 
(1388).  A  frequent  and  free  use  of  the  zero  in  the 
thirteenth  century  is  shown  in  tables  for  the  calcula- 
tion of  the  tides  at  London  and  of  the  duration  of 
moonlight. f  In  the  year  1471  there  appeared  in  Co- 
logne a  work  of  Petrarch  with  page-numbers  in  Hindu 
figures  at  the  top.  In  1482  the  first  German  arith- 
metic with  similar  page-numbering  was  published  in 
Bamberg.  Besides  the  ordinary  forms  of  numerals 
everywhere  used  to-day,  which  appeared  exclusively 
in  an  arithmetic  of  1489,  the  following  forms  for  4,  5, 
7  were  used  in  Germany  at  the  time  of  the  struggle 
between  the  Roman  and  Hindu  notations : 


The  derivation  of  the  modern  numerals  is  illustrated 
by  the  examples  below  which  are  taken  in  succession 
from  the  Sanskrit,  the  apices,  the  Eastern  Arab,  the 

*  linger,  Die  Methodik  der  praktischen  Ariihmetik,  1888,  p.  70.  Hereafter 
referred  to  as  Unger. 

tGunther,  Geschichte  des  mathematischen  Unterrichts  im  deutschen  Mittel- 
alter  bis  zum  Jahr  1525,  1887,  p.  175.  Hereafter  referred  to  as  Gunther. 


NUMBER-SYSTEMS  AND  NUMBER-SYMBOLS.  1  7 

Western  Arab  Gubar  numerals,  the  numerals  of  the 
eleventh,  thirteenth,  and  sixteenth  centuries.* 


dUH  H 

erq'v  8 


5  <^  V  A 

76  A    8 
8 


In  the  sixteenth  century  the  Hindu  position-arith- 
metic and  its  notation  first  found  complete  introduc- 
tion among  all  the  civilized  peoples  of  the  West.  By 
this  means  was  fulfilled  one  of  the  indispensable  con- 
ditions for  the  development  of  common  arithmetic  in 
the  schools  and  in  the  service  of  trade  and  commerce. 

*  Cantor,  table  appended  to  Vol.  I,  and  Hankel,  p.  325. 


II.  ARITHMETIC. 

A.  GENERAL  SURVEY. 

'T^HE  simplest  number- words  and  elementary  count 
*-  ing  have  always  been  the  common  property  of 
the  people.  Quite  otherwise  is  it,  however,  with  the 
different  methods  of  calculation  which  are  derived 
from  simple  counting,  and  with  their  application  to 
complicated  problems.  As  the  centuries  passed,  that 
part  of  ordinary  arithmetic  which  to-day  every  child 
knows,  descended  from  the  closed  circle  of  particular 
castes  or  smaller  communities  to  the  common  people, 
so  as  to  form  an  important  part  of  general  culture. 
Among  the  ancients  the  education  of  the  youth  had  to 
do  almost  wholly  with  bodily  exercises.  Only  a  riper 
age  sought  a  higher  cultivation  through  intercourse 
with  priests  and  philosophers,  and  this  consisted  in 
part  in  the  common  knowledge  of  to-day :  people 
learned  to  read,  to  write,  to  cipher. 

At  the  beginning  of  the  first  period  in  the  historic 
development  of  common  arithmetic  stand  the  Egyp- 
tians. To  them  the  Greek  writers  ascribe  the  inven- 
tion of  surveying,  of  astronomy,  and  of  arithmetic.  To 
their  literature  belongs  also  the  most  ancient  book  on 


ARITHMETIC.  19 

arithmetic,  that  of  Ahmes,  which  teaches  operations 
with  whole  numbers  and  fractions.  The  Babylonians 
employed  a  sexagesimal  system  in  their  position-arith- 
metic, which  latter  must  also  have  served  the  pur- 
poses of  a  religious  number-symbolism.  The  common 
arithmetic  of  the  Greeks,  particularly  in  most  ancient 
times,  was  moderate  in  extent  until  by  the  activity  of 
the  scholars  of  philosophy  there  was  developed  a  real 
mathematical  science  of  predominantly  geometric 
character.  In  spite  of  this,  skill  in  calculation  was 
not  esteemed  lightly.  Of  this  we  have  evidence  when 
Plato  demands  for  his  ideal  state  that  the  youth  should 
be  instructed  in  reading,  writing,  and  arithmetic. 

The  arithmetic  of  the  Romans  had  a  purely  prac- 
tical turn  ;  to  it  belonged  a  mass  of  quite  complicated 
problems  arising  from  controversies  regarding  ques- 
tions of  inheritance,  of  private  property  and  of  reim- 
bursement of  interest.  The  Romans  used  duodecimal 
fractions.  Concerning  the  most  ancient  arithmetic  of 
the  Hindus  only  conjectures  can  be  made  ;  on  the  con- 
trary, the  Hindu  elementary  arithmetic  after  the  in- 
troduction of  the  position-system  is  known  with  toler- 
able accuracy  from  the  works  of  native  authors.  The 
Hindu  mathematicians  laid  the  foundations  for  the 
ordinary  arithmetic  processes  of  to-day.  The  influ- 
ence of  their  learning  is  perceptible  in  the  Chinese 
arithmetic  which  likewise  depends  on  the  decimal  sys- 
tem ;  in  still  greater  measure,  however,  among  the 


2O  HISTORY  OF  MATHEMATICS. 

Arabs  who  besides  the  Hindu  numeral-reckoning  also 
employed  a  calculation  by  columns. 

The  time  from  the  eighth  to  the  beginning  of  the 
fifteenth  century  forms  the  second  period.  This  is  a 
noteworthy  period  of  transition,  an  epoch  of  the  trans- 
planting of  old  methods  into  new  and  fruitful  soil, 
but  also  one  of  combat  between  the  well-tried  Hindu 
methods  and  the  clumsy  and  detailed  arithmetic  ope- 
rations handed  down  from  the  Middle  Ages.  At 
first  only  in  cloisters  and  cloister-schools  could  any 
arithmetic  knowledge  be  found,  and  that  derived  from 
Roman  sources.  But  finally  there  came  new  sugges 
tions  from  the  Arabs,  so  that  from  the  eleventh  to  the 
thirteenth  centuries  there  was  opposed  to  the  group  of 
abacists,  with  their  singular  complementary  methods, 
a  school  of  algorists  as  partisans  of  the  Hindu  arith- 
metic. 

Not  until  the  fifteenth  century,  the  period  of  in- 
vestigation of  the  original  Greek  writings,  of  the 
rapid  development  of  astronomy,  of  the  rise  of  the 
arts  and  of  commercial  relations,  does  the  third  pe- 
riod in  the  history  of  arithmetic  begin.  As  early 
as  the  thirteenth  century  besides  the  cathedral  and 
cloister-schools  which  provided  for  their  own  religious 
and  ecclesiastical  wants,  there  were,  properly  speak- 
ing, schools  for  arithmetic.  Their  foundation  is  to  be 
ascribed  to  the  needs  of  the  brisk  trade  of  German 
towns  with  Italian  merchants  who  were  likewise  skilled 
computers.  In  the  fifteenth  and  sixteenth  centuries 


ARITHMETIC.  21 

school  affairs  were  essentially  advanced  by  the  human- 
istic tendency  and  by  the  reformation.  Latin  schools, 
writing  schools,  German  schools  (in  Germany)  for  boys 
and  even  for  girls,  were  established.  In  the  Latin 
schools  only  the  upper  classes  received  instruction  in 
arithmetic,  in  a  weekly  exercise  :  they  studied  the  four 
fundamental  rules,  the  theory  of  fractions,  and  at  most 
the  rule  of  three,  which  may  not  seem  so  very  little 
when  we  consider  that  frequently  in  the  universities 
of  that  time  arithmetic  was  not  carried  much  further. 
In  the  writing  schools  and  German  boys'  schools  the 
pupils  learned  something  of  calculation,  numeration, 
and  notation,  especially  the  difference  between  the 
German  numerals  (in  Roman  writing)  and  the  ciphers 
(after  the  Hindu  fashion).  In  the  girls'  schools,  which 
were  intended  only  for  the  higher  classes  of  people,  no 
arithmetic  was  taught.  Considerable  attainments  in 
computation  could  be  secured  only  in  the  schools  for 
arithmetic.  The  most  celebrated  of  these  institutions 
was  located  at  Nuremberg.  In  the  commercial  towns 
there  were  accountants'  guilds  which  provided  for  the 
extension  of  arithmetic  knowledge.  But  real  mathe- 
maticians and  astronomers  also  labored  together  in  de- 
veloping the  methods  of  arithmetic.  In  spite  of  this 
assistance  from  men  of  prominence,  no  theory  of  arith- 
metic instruction  had  been  established  even  as  late  as 
in  the  sixteenth  century.  What  had  been  done  be- 
fore had  to  be  copied.  In  the  books  on  arithmetic 


22  HISTORY  OF  MATHEMATICS. 

were  found  only  rules  and  examples,   almost  never 
proofs  or  deductions. 

The  seventeenth  century  brought  no  essential 
change  in  these  conditions.  Schools  existed  as  before 
where  they  had  not  been  swallowed  up  by  the  horrors 
of  the  Thirty- Years'  War.  The  arithmeticians  wrote 
their  books  on  arithmetic,  perhaps  contrived  calculat- 
ing machines  to  make  the  work  easier  for  their  pupils, 
or  composed  arithmetic  conversations  and  poems.  A 
specimen  of  this  is  given  in  the  following  extracts 
from  Tobias  Beutel's  Arithmetica,  the  seventh  edition 
of  which  appeared  in  1693.* 

"  Numerieren  lehrt  im  Rechen 
Zahlen  schreiben  und  aussprechen." 

"In  Summen  bringen  heisst  addieren 
Dies  muss  das  Wdrtlein  Und  vollfiihren." 

"  Wie  eine  Hand  an  uns  die  andre  waschet  rein 
Kann  eine  Species  der  andern  Probe  seyn." 

"  We  are  taught  in  numeration 
Number  writing  and  expression," 
etc.,  etc. 

Commercial  arithmetic  was  improved  by  the  cultiva 
tion  of  the  study  of  exchange  and  discount,  and  the 
abbreviated  method  of  multiplication.  The  form  of 
instruction  remained  the  same,  i.  e.,  the  pupil  reck- 
oned according  to  rules  without  any  attempt  being 
made  to  explain  their  nature. 

The  eighteenth  century  brought  as  its  first  and 

*  UnRer,  p.  124. 


ARITHMETIC.  23 

most  important  innovation  the  statutory  regulation  of 
school  matters  by  special  school  laws,  and  the  estab- 
lishment of  normal  schools  (the  first  in  1732  at  Stet- 
tin in  connection  with  the  orphan  asylum).  As  reor- 
ganizers  of  the  higher  schools  appeared  the  pietists 
and  philanthropinists.  The  former  established  Real- 
schulen  (the  oldest  founded  1738  in  Halle)  and  higher 
Biirgerschulen;  the  latter  in  their  Schulen  der  Aufkldrung 
sought  by  an  improvement  of  methods  to  educate 
cultured  men  of  the  world.  The  arithmetic  exercise- 
books  of  this  period  contain  a  simplification  of  divi- 
sion (the  downwards  or  under-itself  division)  as  well 
as  a  more  fruitful  application  of  the  chain  rule  and 
decimal  fractions.  By  their  side  also  appear  manuals 
of  method  whose  number  is  rapidly  increasing  in  the 
nineteenth  century.  In  these,  elementary  teaching 
receives  especial  attention.  According  to  Pestalozzi 
(1803)  the  foundation  of  calculation  is  sense  percep- 
tion, according  to  Grube  (1842),  the  comprehensive 
treatment  of  each  number  before  taking  up  the  next, 
according  to  Tanck  and  Knilling  (1884),  counting. 
In  Pestalozzi's  method  "the  decimal  structure  of  our 
number-system,  which  includes  so  many  advantages 
in  the  way  of  calculation,  is  not  touched  upon  at  all, 
addition,  subtraction,  and  division  do  not  appear  as 
separate  processes,  the  accompanying  explanations 
smother  the  principal  matter  in  the  propositions,  that 
is  the  arithmetic  truth."*  Grube  has  simply  drawn 

*  Unger,  p.  179. 


24  HISTORY  OF  MATHEMATICS. 

from  Pestalozzi's  principles  the  most  extreme  conclu 
sions.  His  sequence  "is  in  many  respects  faulty;  his 
processes  unsuitable."*  The  historical  development 
of  arithmetic  speaks  in  favor  of  the  counting-prin- 
ciple :  the  first  reckoning  in  every  age  has  been  an 
observing  and  counting. 

B.  FIRST  PERIOD. 

THE    ARITHMETIC    OF    THE    OLDEST    NATIONS  TO  THE  TIME 
OF  THE  ARABS. 

I.    The  Arithmetic  of  Whole  Numbers. 

If  we  leave  out  of  account  finger-reckoning,  which 
cannot  be  shown  with  absolute  certainty,  then  accord- 
ing to  a  statement  of  Herodotus  the  ancient  Egyptian 
computation  consisted  of  an  operating  with  pebbleson 
a  reckoning-board  whose  lines  were  at  right  angles  to 
the  computer.  Possibly  the  Babylonians  also  used  a 
similar  device.  In  the  ordinary  arithmetic  of  the  latter, 
as  among  the  Egyptians,  the  decimal  system  prevails, 
but  by  its  side  we  also  find,  especially  in  dealing  with 
fractions,  a  sexagesimal  system.  This  arose  without 
doubt  in  the  working  out  of  the  astronomical  observa- 
tions of  the  Babylonian  priests,  f  The  length  of  the 
year  of  360  days  furnished  the  occasion  for  the  divi- 
sion of  the  circle  into  360  equal  parts,  one  of  which 
was  to  represent  the  apparent  daily  path  of  the  sun 
upon  the  celestial  sphere.  If  in  addition  the  construc- 

*Unger,  pp.  192,  193.  t Cantor,  I.,  p.  80. 


ARITHMETIC.  25 

tion  of  the  regular  hexagon  was  known,  then  it  was 
natural  to  take  every  60  of  these  parts  again  as  units. 
The  number  60  was  called  soss.  Numbers  of  the 
sexagesimal  system  were  again  multiplied  in  accord- 
ance with  the  rules  of  the  decimal  system  :  thus  a  ner 
=  600,  a  jar  =  3600.  The  sexagesimal  system  estab- 
lished by  the  Babylonian  priests  also  entered  into 
their  religious  speculations,  where  each  of  their  divin- 
ities was  designated  by  one  of  the  numbers  from  1  to 
60  corresponding  to  his  rank.  Perhaps  the  Babyloni- 
ans also  divided  their  days  into  60  equal  parts  as  has 
been  shown  for  the  Veda  calendars  of  the  ancient 
Hindus. 

The  Greek  elementary  mathematics,  at  any  rate 
as  early  as  the  time  of  Aristophanes  (420  B.  C.),*  used 
finger-reckoning  and  reckoning-boards  for  ordinary 
computation.  An  explanation  of  the  finger-reckoning 
is  given  by  Nicholas  Rhabdaf  of  Smyrna  (in  the  four- 
teenth century).  Moving  from  the  little  finger  of  the 
left  hand  to  the  little  finger  of  the  right,  three  fingers 
were  used  to  represent  units,  the  next  two,  tens,  the 
next  two,  hundreds,  and  the  last  three,  thousands. 
On  the  reckoning  board,  the  abax  (5/?o£,  dust  board), 
whose  columns  were  at  right  angles  to  the  user,  the 
operations  were  carried  on  with  pebbles  which  had  a 
different  place-value  in  different  lines.  Multiplication 
was  performed  by  beginning  with  the  highest  order  in 
each  factor  and  forming  the  sum  of  the  partial  pro- 

*  Cantor,  I ..  pp.  120, 479.        t  Gow,  History  of  Greek  Mathematics,  p.  24. 


26 


HISTORY  OF  MATHEMATICS. 


ducts.     Thus  the  calculation  was  effected  (in  modern 
form)  as  follows: 

126  •  237  =  (100  +  20  +  6)  (200  +  30  +  7) 
=       20000+     3000          +700 
+     4000+        600          +140 
+     1200+        180          +   42 

=       29  862 

According  to  Pliny,  the  finger-reckoning  of  the 
Romans  goes  back  to  King  Numa ;  *  the  latter  had 
made  a  statue  of  Janus  whose  fingers  represented  the 
number  of  the  days  of  a  year  (355).  Consistently  with 
this  Boethius  calls  the  numbers  from  1  to  9  finger- 
numbers,  10,  20,  30,  ...  joint-numbers,  11,  12,  ... 
19,  21,  22,  ...  29,  ...  composite  numbers.  In  ele 

f  n  1 1  n  1 1 

[Xl  ©   0    *    C    X    I    9    | 


\ 


mentary   teaching    the    Romans   used    the   abacus,   a 
board  usually  covered  with  dust  on  which  one  could 


*  Cantor,  I.,  p.  491. 


ARITHMETIC.  2J 

trace  figures,  draw  columns,  and  work  with  pebbles. 
Or  if  the  abacus  was  to  be  used  for  computing  only, 
it  was  made  of  metal  and  provided  with  grooves  (the 
vertical  lines  in  the  schematic  drawing  on  the  pre- 
ceding page)  in  which  arbitrary  marks  (the  cross- 
lines)  could  be  shifted. 

The  columns  a\  .  .  .  aj,  b\  .  .  .  bi  form  a  system 
from  1  to  1  000  000 ;  upon  a  column  a  are  found  four 
marks,  upon  a  column  b  only  one  mark.  Each  of  the 
four  marks  represents  a  unit,  but  the  upper  single 
mark  five  units  of  the  order  under  consideration. 
Further  a  mark  upon  fi=^y,  upon  £*  =  &,  upon  4 
=  A>  uPon  f*  =  ^s>  upon  cs  =  -fa  (relative  to  the  di- 
vision of  the  a's).  The  abacus  of  the  figure  represents 
the  number  782  192  +  ^  +  ^  +  ^  =  782  192 1 1.  This" 
abacus  served  for  the  reckoning  of  results  of  simple 
problems.  Along  with  this  the  multiplication-table 
was  also  employed.  For  larger  multiplications  there 
were  special  tables.  Such  a  one  is  mentioned  by  Vic- 
torius  (about  450  A.  D.).*  From  Boethius,  who  calls 
the  abacus  marks  apices,  we  learn  something  about 
multiplication  and  division.  Of  these  operations  the 
former  probably,  the  latter  certainly,  was  performed 
by  the  use  of  complements.  In  Boethius  the  term 
differentia  is  applied  to  the  complement  of  the  divisor 
to  the  next  complete  ten  or  hundred.  Thus  for  the 
divisors  7,  84,  213  the  differentiae  are  3,  6,  87  f  respec- 
tively. The  essential  characteristics  of  this  comple- 

*  Cantor,  I.,  p.  495.  t  Cantor,  I ,  p.  544. 


28  HISTORY  OF  MATHEMATICS. 

mentary  division  are  seen  from  the  following  example 
put  in  modern  form  : 
257 


14    20  —  6 

_lU-f- 

20  —  6 

1  20  —  6 

117 

=  5  + 

30  +  17 

5  1   47 

20—6 

20  —  6  ~ 

1  20—6 

47 
20^6 

-  2  + 

12  +  7 

2  1   19 

20  —  6  ~ 

1  20-6 

19 

5 

14 

=  1  + 

H 

257 

18  + 

5 
14* 

The  swanpan  of  the  Chinese  somewhat  resembles 
the  abacus  of  the  Romans.  This  calculating  machine 
consists  of  a  frame  ordinarily  with  ten  wires  inserted. 
A  cross  wire  separates  each  of  the  ten  wires  into  two 
unequal  p^arts ;  on  each  smaller  part  two  and  on  each 
larger  five  balls  are  strung.  The  Chinese  arithmetics 
give  no  rules  for  addition  and  subtraction,  but  do  for 
multiplication,  which,  as  with  the  Greeks,  begins 
with  the  highest  order,  and  fordivision,  which  appears 
in  the  form  of  a  repeated  subtraction. 

The  calculation  of  the  Hindus,  after  the  introduc- 
tion of  the  arithmetic  of  position,  possessed  a  series 
of  suitable  rules  for  performing  the  fundamental  ope- 
rations. In  the  case  of  a  smaller  figure  in  the  minu- 
end subtraction  is  performed  by  borrowing  and  by 
addition  (as  in  the  so-called  Austrian  subtraction).* 

*The  Austrian  subtraction  corresponds  in  part  to  the  usual  method  of 
"making  change." 


ARITHMETIC. 


29 


In  multiplication,  for  which  several  processes  are 
available,  the  product  is  obtained  in  some  cases 
by  separating  the  multipliers  into  factors  and  subse- 
quently adding  the  partial  products.  In  other  cases 
a  schematic  process  is  introduced  whose  peculiarities 
are  shown  in  the  example  315-37  =  11  655. 

1         6 


The  result  of  the  multiplication  is  obtained  by  the 
addition  of  the  figures  found  within  the  rectangle  in 
the  direction  of  the  oblique  lines.  With  regard  to 
division  we  have  only  a  few  notices.  Probably,  how- 
ever, complementary  methods  were  not  used. 

The  earliest  writer  giving  us  information  on  the 
arithmetic  of  the  Arabs  is  Al  Khowarazmi.  The  bor- 
rowing from  Hindu  arithmetic  stands  out  very  clearly. 
Six  operations  were  taught.  Addition  and  subtraction 
begin  with  the  units  of  highest  order,  therefore  on 
the  left ;  halving  begins  on  the  right,  doubling  again 
on  the  left.  Multiplication  is  effected  by  the  process 
which  the  Hindus  called  Tatstha  (it  remains  stand- 
ing).* The  partial  products,  beginning  with  the  high- 
est order  in  the  multiplicand,  are  written  above  the 
corresponding  figures  of  the  latter  and  each  figure 

*  Cantor,  I.,  p.  674,  571. 


30  HISTORY  OF  MATHEMATICS. 

of  the  product  to  which  other  units  from  a  later  par- 
tial product  are  added  (in  sand  or  dust),  rubbed  out 
and  corrected,  so  that  at  the  end  of  the  computation 
the  result  stands  above  the  multiplicand.  In  divi- 
sion, which  is  never  performed  in  the  complementary 
fashion,  the  divisor  stands  below  the  dividend  and 
advances  toward  the  right  as  the  calculation  goes  on. 
Quotient  and  remainder  appear  above  the  divisor  in 
^L  =  28£f,  somewhat  as  follows:* 

13 
14 
28 
461 
16 

16 

Al  Nasawif  also  computes  after  the  same  fashion  as 
Al  Khowarazmi.  Their  methods  characterise  the  ele- 
mentary arithmetic  of  the  Eastern  Arabs. 

In  essentially  the  same  manner,  but  with  more  or 
less  deviation  in  the  actual  work,  the  Western  Arabs 
computed.  Besides  the  Hindu  figure-computation 
Ibn  al  Banna  teaches  a  sort  of  reckoning  by  columns.  J 
Proceeding  from  right  to  left,  the  columns  are  com- 
bined in  groups  of  three;  such  a  group  is  called  ta- 
karrur\  the  number  of  all  the  columns  necessary  to 
record  a  number  is  the  mukarrar.  Thus  for  the  num- 
ber 3  849  922  the  takarrur  or  number  of  complete 
groups  is  2,  the  mukarrar -=1 .  Al  Kalsadi  wrote  a 

*  Cantor,  I.,  p.  674.  t  Cantor,  I.,  p.  716.  t  Cantor,  I.,  p.  757. 


ARITHMETIC.  31 

work  Raising  of  the  Veil  of  the  Science  of  Gubar.  *  The 
original  meaning  of  Gubar  (dust)  has  here  passed 
over  into  that  of  the  written  calculation  with  figures. 
Especially  characteristic  is  it  that  in  addition,  sub- 
traction (=tarh,  taraha  =  to  throw  away)  and  multi- 
plication the  results  are  written  above  the  numbers 
operated  upon,  as  in  the  following  examples : 

1 93  +  45  =  238  and  238  — 193  =  45 

is  written,  is  written, 

238  45 

193 ;  238' 

45  193 

1  1 

Several  rules  for  multiplication  are  found  in  Al  Kal- 
sadi,  among  them  one  with  an  advancing  multiplier. 
In  division  the  result  stands  below. 

FIRST  EXAMPLE.  SECOND  EXAMPLE. 

7-143  =  1001  1001  _ 

is  written,  1001  7 

21  is  written,  32 

28  1001 

7  777 

~T43  143 

777 

2.    Calculation  With  Fractions. 

In  his  arithmetic  Ahmes  gives  a  large  number  of 
examples  which  show  how  the  Egyptians  dealt  with 
fractions.  They  made  exclusive  use  of  unit-fractions, 

*  Cantor,  I.,  p.  762. 


32  HISTORY  OF  MATHEMATICS. 

i.  e.,  fractions  with  numerator  1.  For  this  numerator, 
therefore,  a  special  symbol  is  found,  in  the  hiero- 
glyphic writing  o,  in  the  hieratic  a  point,  so  that  in 
the  latter  a  unit  fraction  is  represented  by  its  denomi- 
nator with  a  point  placed  above  it.  Besides  these 
there  are  found  for  £  and  f  the  hieroglyphs  I  and 

jj) ;  *  in  the  hieratic  writing  there  are  likewise  special 
symbols  corresponding  to  the  fractions  £,  f,  ^,  and  i. 
The  first  problem  which  Ahmes  solves  is  this,  to  sep- 
arate a  fraction  into  unit  fractions.  E.  g.,  he  finds 
l  =  i  +  T^'  inr  =  TrV  +  Tmr  +  i>7ir  This  separation, 
really  an  indeterminate  problem,  is  not  solved  by 
Ahmes  in  general  form,  but  only  for  special  cases. 

The  fractions  of  the  Babylonians  being  entirely 
in  the  sexagesimal  system,  had  at  the  outset  a  com- 
mon denominator,  and  could  be  dealt  with  like  whole 
numbers.  In  the  written  form  only  the  numerator 
was  given  with  a  special  sign  attached.  The  Greeks 
wrote  a  fraction  so  that  the  numerator  came  first  with 
a  single  stroke  at  the  right  and  above,  followed  in  the 
same  line  by  the  denominator  with  two  strokes,  writ- 
ten twice,  thus  i£'ica"Ka"  =  ^J.  In  unit  fractions  the 
numerator  was  omitted  and  the  denominator  written 
only  once:  8"  =  £.  The  unit  fractions  to  be  added 
follow  immediately  one  after  another,  f  £"  107"  pip"  o-xS" 
=  $H-*V  +  Tiir  + A=s4sV  In  arithmetic  proper, 
extensive  use  was  made  of  unit-fractions,  later  also  of 

*For  carefully  drawn  symbols  see  Cantor,  I.  p.  45. 
t  Cantor,  I.,  p.  118. 


ARITHMETIC.  33 

sexagesimal  fractions  (in  the  computation  of  angles). 
Of  the  use  of  a  bar  between  the  terms  of  a  fraction 
there  is  nowhere  any  mention.  Indeed,  where  such 
use  appears  to  occur,  it  marks  only  the  result  of  an 
addition,  but  not  a  division.* 

The  fractional  calculations  of  the  Romans  furnish 
an  example  of  the  use  of  the  duodecimal  system. 
The  fractions  (minutice)  -fa,  ^,  .  .  .  \$  had  special 
names  and  symbols.  The  exclusive  use  of  these  duo- 
decimal fractions  f  was  due  to  the  fact  that  the  as, 
a  mass  of  copper  weighing  one  pound,  was  divided 
into  twelve  uncice.  The  uncia  had  four  sicilici  and 
twenty-  four  scripuli.  I=as,  % = semis,  §  =  trfcns,  £  = 
quadrans,  etc.  Besides  the  twelfths  special  names 
were  given  to  the  fractions  fa  -fa,  ^,  j^,  ^.  The 
addition  and  subtraction  of  such  fractions  was  com- 
paratively simple,  but  their  multiplication  very  de- 
tailed. The  greatest  disadvantage  of  this  system  con- 
sisted in  the  fact  that  all  divisions  which  did  not  fit 
into  this  duodecimal  system  could  be  represented  by 
minutiae  either  with  extreme  difficulty  or  only  approxi- 
mately. 

In  the  computations  of  the  Hindus  both  unit  frac- 
tions and  derived  fractions  likewise  appear.  The  de- 
nominator stands  under  the  numerator  but  is  not  sep- 
arated from  it  by  a  bar.  The  Hindu  astronomers 
preferred  to  calculate  with  sexagesimal  fractions.  In 
the  computations  of  the  Arabs  Al  Khowarazmi  gives 

*Tmnnery  in  Bibl.  Math.    1886.  tHankel,  p.  57. 


34  HISTORY  OF  MATHEMATICS. 

special  words  for  half,  third,  .  .  .  ninth  (expressible 
fractions).*  All  fractions  with  denominators  non-divis- 
ible by  2,  3,  ...  9,  are  called  mute  fractions ;  they 
were  expressed  by  a  circumlocution,  e.  g.,  ^  as  2 
parts  of  17  parts.  Al  Nasawi  writes  mixed  numbers 
in  three  lines,  one  under  another,  at  the  top  the  whole 
number,  below  this  the  numerator,  below  this  the  de- 
nominator. For  astronomical  calculations  fractions 
of  the  sexagesimal  system  were  used  exclusively. 

3.   Applied  Arithmetic. 

The  practical  arithmetic  of  the  ancients  included 
besides  the  common  cases  of  daily  life,  astronomical 
and  geometrical  problems.  The  latter  will  be  passed 
over  here  because  they  are  mentioned  elsewhere.  In 
Ahmes  problems  in  partnership  are  developed  and 
also  the  sums  of  some  of  the  simplest  series  deter- 
mined. Theon  of  Alexandria  showed  how  to  obtain 
approximately  the  square  root  of  a  number  of  angle 
degrees  by  the  use  of  sexagesimal  fractions  and  the 
gnomon.  The  Romans  were  concerned  principally 
with  problems  of  interest  and  inheritance.  The  Hin- 
dus had  already  developed  the  method  of  false  posi- 
tion (Regula  falsi')  and  the  rule  of  three,  and  made 
a  study  of  problems  of  alligation,  cistern-filling,  and 
series,  which  were  still  further  developed  by  the  Arabs. 
Along  with  the  practical  arithmetic  appear  frequent 

•Cantor,  I.,  p.  675. 


ARITHMETIC.  35 

traces  of  observations  on  the  theory  of  numbers.  The 
Egyptians  knew  the  test  of  divisibility  of  a  number  by 
2.  The  Pythagoreans  distinguished  numbers  as  odd 
and  even,  amicable,  perfect,  redundant  and  defective.* 
Of  two  amicable  numbers  each  was  equal  to  the  sum 
of  the  aliquot  parts  of  the  other  (220  gives  1  +  2  +  4 
+  5  +  10+11  +  20  +  22  +  44  +  55  +  110  =  284  and 
284  gives  1  +  2  +  4  +  71  +  142  =  220).  A  perfect  num- 
ber was  equal  to  the  sum  of  its  aliquot  parts  (6  =  1  + 
2  +  3).  If  the  sum  of  the  aliquot  parts  was  greater  or 
less  than  the  number  itself,  then  the  latter  was  called 
redundant  or  defective  respectively  (8  >  1  +  2  +  4 ;  12 
<l  +  2  +  3  +  4+6).  Besides  this,  Euclid  starting 
from  his  geometric  standpoint  commenced  some  fun- 
damental investigations  on  divisibility,  the  greatest 
common  measure  and  the  least  common  multiple. 
The  Hindus  were  familiar  with  casting  out  the  nines 
and  with  continued  fractions,  and  from  them  this 
knowledge  went  over  to  the  Arabs.  However  insig- 
nificant may  be  these  beginnings  in  their  ancient 
form,  they  contain  the  germ  of  that  vast  development 
in  the  theory  of  numbers  which  the  nineteenth  cen- 
tury has  brought  about. 

*  Cantor,  I.,  p.  156. 


36  HISTORY  OF  MATHEMATICS. 

C.    SECOND  PERIOD. 

FROM  THE  EIGHTH  TO  THE  FOURTEENTH  CENTURY. 

I.    The  Arithmetic  of  Whole  Numbers. 

In  the  cloister  schools,  the  episcopal  schools,  and 
the  private  schools  of  the  Merovingian  and  Carloving- 
ian  period  it  was  the  monks  almost  exclusively  who 
gave  instruction.  The  cloister  schools  proper  were  of 
only  slight  importance  in  the  advancement  of  mathe- 
matical knowledge  :  on  the  contrary,  the  episcopal 
and  private  schools,  the  latter  based  on  Italian  meth- 
ods, seem  to  have  brought  very  beneficial  results. 
The  first  to  foreshadow  something  of  the  mathemat- 
ical knowledge  of  the  monks  is  Isidorus  of  Seville. 
This  cloister  scholar  confined  himself  to  making  con- 
jectures regarding  the  derivation  of  the  Roman  nu- 
merals, and  says  nothing  at  all  about  the  method  of 
computation  of  his  contemporaries.  The  Venerable 
Bede  likewise  published  only  some  extended  observa- 
tions on  finger-reckoning.  He  shows  how  to  repre 
sent  numbers  by  the  aid  of  the  fingers,  proceeding 
from  left  to  right,  and  thereby  assumes  a  certain  ac- 
quaintance with  finger-reckoning,  mentioning  as  his 
predecessors  Macrobius  and  Isidorus.*  This  calculus 
digitalis,  appearing  in  both  the  East  and  the  West  in 

*  Cantor,  I.,  p.  778. 


ARITHMETIC.  37 

exactly  the  same  fashion,  played  an  important  part  in 
fixing  the  dates  of  church  feasts  by  the  priests  of  that 
time ;  at  least  computus  digitalis  and  computus  ecclesias- 
ticus  were  frequently  used  in  the  same  sense.* 

With  regard  to  the  fundamental  operations  proper 
Bede  does  not  express  himself.  Alcuin  makes  much 
of  number-mysticism  and  reckons  in  a  very  cumbrous 
manner  with  the  Roman  numerals,  f  Gerbert  was  the 
first  to  give  in  his  Regula  de  abaco  computi  actual  rules, 
in  which  he  depended  upon  the  arithmetic  part  of 
Boethius's  work.  What  he  teaches  is  a  pure  abacus- 
reckoning,  which  was  widely  spread  by  reason  of  his 
reputation.  Gerbert's  abacus,  of  which  we  have  an 
accurate  description  by  his  pupil  Bernelinus,  was  a 
table  which  for  the  drawing  of  geometric  figures  was 
sprinkled  with  blue  sand,  but  for  calculation  was  di- 
vided into  thirty  columns  of  which  three  were  reserved 
for  fractional  computations.  The  remaining  twenty- 
seven  columns  were  separated  from  right  to  left  into 
groups  of  three.  At  the  head  of  each  group  stood  like- 
wise from  right  to  left  S  (singularis} ,  D  (decent),  C  (cen- 
tum). The  number-symbols  used,  the  so-called  apices, 
are  symbols  for  1  to  9,  but  without  zero.  In  calcu- 
lating with  this  abacus  the  intermediate  operations 
could  be  rubbed  out,  so  that  finally  only  the  result  re- 
mained ;  or  the  operation  was  made  with  counters. 
The  fundamental  operations  were  performed  princi- 
pally by  the  use  of  complements,  and  in  this  respect 

*  Giinther.  t  Camber. 


HISTORY  OF  MATHEMATICS. 


division  is  especially  characteristic.  The  formation 
of  the  quotient  if ^  =  33^  will  explain  this  comple- 
mentary division. 


c 

D 

S 

4 

1 

9 

9 

1. 

9. 

9. 

4. 

1. 

6. 

1. 

4. 

4 

9. 

1. 

6. 

4. 

1. 

9. 

4. 

1. 

3. 

4. 

7. 

1 

1. 

4. 

1. 

1. 

4. 

1. 

1. 

1. 

1. 

3 

3 

C 

D 

S 
6 

1 

9 

9 

1. 

9. 

9. 

1 

3 

3 

10  —  4 

199 

99+40 
139 

10 

10 

7 

3 
1 
1 
1 

79 

9+28 

37 

19 

13 

7 

1 

33 

In  the  example  given  the  complete  performance  of  the  com- 
plementary division  stands  on  the  left ;  the  figures  to  be  rubbed 
out  as  the  calculation  goes  on  are  indicated  by  a  period  on  the 
right.  On  the  right  is  found  the  abacus-division  without  the  for- 
mation of  the  difference  in  the  divisor,  below  it  the  explanation  of 
the  complementary  division  in  modern  notation. 


ARITHMETIC.  39 

In  the  tenth  and  eleventh  centuries  there  appeared 
a  large  number  of  authors  belonging  chiefly  to  the 
clergy  who  wrote  on  abacus-reckoning  with  apices 
but  without  the  zero  and  without  the  Hindu-Arab 
methods.  In  the  latter  the  apices  were  connected  with 
the  abacus  itself  or  with  the  representation  of  num- 
bers of  one  figure,  while  in  the  running  text  the  Roman 
numeral  symbols  stood  for  numbers  of  several  figures. 
The  contrast  between  the  apices-plan  and  the  Roman 
is  so  striking  that  Oddo,  for  example,  writes  :  "If  one 
takes  5  times  7,  or  7  times  5,  he  gets  XXXV"  (the  5 
and  7  written  in  apices).* 

At  the  time  of  the  abacus-reckoning  there  arose  the  peculiar 
custom  of  representing  by  special  signs  certain  numbers  which  do 
not  appear  in  the  Roman  system  of  symbols,  and  this  use  contin- 
ued far  into  the  Middle  Ages.  Thus,  for  example,  in  the  town- 
books  of  Greifswald  250  is  continually  represented  by 


The  abacists  with  their  remarkable  methods  of  di- 
vision completely  dominated  Western  reckoning  up 
to  the  beginning  of  the  twelfth  century.  But  then  a 
complete  revolution  was  effected.  The  abacus,  the 
heir  of  the  computus,  i.  e.,  the  old  Roman  method  of 
calculation  and  number-writing,  was  destined  to  give 
way  to  the  algorism  with  its  sensible  use  of  zero  and 
its  simpler  processes  of  reckoning,  but  not  without  a 
further  struggle.  J  People  became  pupils  of  the  Wes- 
tern Arabs.  Among  the  names  of  those  who  extended 

*  Cantor,  I.,  p.  846.  t  Gunther,  p.  175.  t  Giinther,  p.  107. 


40  HISTORY  OF  MATHEMATICS. 

Arab  methods  of  calculation  stands  forth  especially 
pre-eminent  that  of  Gerhard  of  Cremona,  because  he 
translated  into  Latin  a  series  of  writings  of  Greek 
and  Arab  authors.*  Then  was  formed  the  school  of 
algorists  who  in  contrast  to  the  abacists  possessed  no 
complementary  division  but  did  possess  the  Hindu 
place-system  with  zero.  The  most  lasting  material 
for  the  extension  of  Hindu  methods  was  furnished  by 
Fibonacci  in  his  Liber  abaci.  This  book  "has  been 
the  mine  from  which  arithmeticians  and  algebraists 
have  drawn  their  wisdom  ;  on  this  account  it  has  be- 
come in  general  the  foundation  of  modern  science."! 
Among  other  things  it  contains  the  four  rules  for 
whole  numbers  and  fractions  in  detailed  form.  It  is 
worthy  of  especial  notice  that  besides  ordinary  sub- 
traction with  borrowing  he  teaches  subtraction  by  in- 
creasing the  next  figure  of  the  subtrahend  by  one, 
and  that  therefore  Fibonacci  is  to  be  regarded  as  the 
creator  of  this  elegant  method. 

2.  Arithmetic  of  Fractions. 

Here,  also,  after  Roman  duodecimal  fractions  had 
been  exclusively  cultivated  by  the  abacists  Beda,  Ger- 
bert  and  Bernelinus,  Fibonacci  laid  a  new  foundation 
in  his  exercises  preliminary  to  division.  He  showed 
how  to  separate  a  fraction  into  unit  fractions.  Espe- 
cially advantageous  in  dealing  with  small  numbers 

*  Hankel,  p.  336.  t  Hankel,  p.  343. 


ARITHMETIC.  41 

is  his  method  of  determining  the  common  denomina- 
tor: the  greatest  denominator  is  multiplied  by  each 
following  denominator  and  the  greatest  common  meas- 
ure of  each  pair  of  factors  rejected.  (Example  :  the 
least  common  multiple  of  24,  18,  15,  9,  8,  5  is  24-3-5 
=  360.) 

j.  Applied  Arithmetic. 

The  arithmetic  of  the  abacists  had  for  its  main 
purpose  the  determination  of  the  date  of  Easter.  Be- 
sides this  are  found,  apparently  written  by  Alcuin, 
Problems  for  Quickening  the  Mind  which  suggest  Ro- 
man models.  In  this  department  also  Leonardo  Fibo- 
nacci furnishes  the  most  prominent  rule  (the  regula 
falsi),  but  his  problems  belong  more  to  the  domain  of 
algebra  than  to  that  of  lower  arithmetic. 

Investigations  in  the  theory  of  numbers  could 
hardly  be  expected  from  the  school  of  abacists.  On 
the  other  hand,  the  algorist  Leonardo  was  familiar 
with  casting  out  the  nines,  for  which  he  furnished  an 
independent  proof. 


D.    THIRD  PERIOD. 

FROM  THE  FIFTEENTH  TO  THE  NINETEENTH  CENTURY. 

I.    The  Arithmetic  of  Whole  Numbers. 

While  on  the  whole  the  fourteenth  century  had 
only  reproductions  to  show,  a  new  period  of  brisk  ac- 


42  HISTORY  OF  MATHEMATICS. 

tivity  begins  with  the  fifteenth  century,  marked  by 
Peurbach  and  Regiomontanus  in  Germany,  and  by 
Luca  Pacioli  in  Italy.  As  far  as  the  individual  pro- 
cesses are  concerned,  in  addition  the  sum  sometimes 
stands  above  the  addends,  sometimes  below;  subtrac- 
tion recognizes  "carrying"  and  "borrowing";  in 
multiplication  various  methods  prevail ;  in  division  no 
settled  method  is  yet  developed.  The  algorism  of 
Peurbach  names  the  following  arithmetic  operations  : 
Numeratio,  additio,  subtractio,  mediatio,  duplatio,  multi- 
plicatio,  divisio,  progressio  (arithmetic  and  geometric 
series),  besides  the  extraction  of  roots  which  before  the 
invention  of  decimal  fractions  was  performed  by  the 
aid  of  sexagesimal  fractions.  His  upwards-division 
still  used  the  arrangement  of  the  advancing  divisor ; 
it  was  performed  in  the  manner  following  (on  the  left 
the  explanation  of  the  process,  on  the  right  Peurbach's 
division,  where  figures  to  be  erased  in  the  course  of 
the  reckoning  are  indicated  by  a  period  to  the  right 
and  below):  The  oral  statement  would  be  somewhat 
like  this:  36  in  84  twice,  2-3  =  6,  8  —  6  =  2,  written 
above  8;  2-6  =  12,  24  — 12  =  12,  write  above,  strike 
out  2,  etc.  The  proof  of  the  accuracy  of  the  result  is 
obtained  as  in  the  other  operations  by  casting  out  the 
nines.  This  method  of  upwards-division  which  is  not 
difficult  in  oral  presentation  is  still  found  in  arith- 
metics which  appeared  shortly  before  the  beginning 
of  the  nineteenth  century. 


ARITHMETIC.  43 


8479|235 
6 

24~ 
12 

12  1-1 

1.3.4. 
-07-  2.2.9.9 

8  4  7  9  I  235 

J±L  3666 

19 
15 

~49 
30 


In  the  sixteenth  century  work  in  arithmetic  had 
entered  the  Latin  schools  to  a  considerable  extent ;  but 
to  the  great  mass  of  children  of  the  common  people 
neither  school  men  nor  statesmen  gave  any  thought 
before  1525.  The  first  regulation  of  any  value  in  this 
line  is  the  Bavarian  Schuelordnungk  de  anno  1348  which 
introduced  arithmetic  as  a  required  study  into  the  vil- 
lage schools.  Aside  from  an  occasional  use  of  finger- 
reckoning,  this  computation  was  either  a  computation 
upon  lines  with  counters  or  a  figure- computation.  In 
both  cases  the  work  began  with  practice  in  numeration 
in  figures.  To  perform  an  operation  with  counters  a 
series  of  horizontal  parallels  was  drawn  upon  a  suit- 
able base.  Reckoned  from  below  upward  each  counter 
upon  the  1st,  2d,  3d,  .  .  .  line  represented  the  value 
1,  10,  100,  .  .  .,  but  between  the  lines  they  represented 
5,  50,  500,  .  .  .  The  following  figure  shows  the  rep- 


44  HISTORY  OF  MATHEMATICS. 

resentation  of  41  096£.  In  subtraction  the  minuend, 
in  multiplication  the  multiplicand  was  put  upon  the 
lines.  Division  was  treated  as  repeated  subtractions. 
This  line-reckoning  was  completely  lost  in  the  seven- 


O     O     O     O 


©Q  fN  f^N  /^ 

yr-v  \J        vy        \~/ 


~0 

teenth  century  when  it  gave  place  to  real  written 
arithmetic  or  figure-reckoning  by  which  it  had  been 
accompanied  in  the  better  schools  almost  from  the 
first. 

In  the  ordinary  business  and  trade  of  the  Middle 
Ages  use  was  also  made  of  the  widely-extended  score- 
reckoning.  At  the  beginning  of  the  fifteenth  century 
this  method  was  quite  usual  in  Frankfort  on  the  Main, 
and  in  England  it  held  its  own  even  into  the  nine- 
teenth century.  Whenever  goods  were  bought  of  a 
merchant  on  credit  the  amount  was  represented  by 
notches  cut  upon  a  stick  which  was  split  in  two  length- 
wise so  that  of  the  two  parts  which  matched,  the  debtor 
kept  one  and  the  creditor  one  so  that  both  were  se 
cured  against  fraud.* 

In  the  cipher-reckoning  the  computers  of  the  six- 
teenth century  generally  distinguished  more  than  4 
operations;  some  counted  9,  i.  e.,  the  8  named  by 

*  Cantor,  M.    Mathem.  Beitr.  turn  Kulturleben  der  VSlker.     Halle,  1863. 


ARITHMETIC.  45 

Peurbach  and  besides,  as  a  ninth  operation,  evolution, 
the  extraction  of  the  square  root  by  the  formula  (a-\-b}* 
=  a? -\-Zab -\-P,  and  the  extraction  of  the  cube  root 
by  the  formula  (a  +  £)8  =  a8  +  (a  +  £)  Zab  +  P.  Defi- 
nitions appeared,  but  these  were  only  repeated  circum- 
locutions. Thus  Grammateus  says  :  "Multiplication 
shows  how  to  multiply  one  number  by  the  other. 
Subtraction  explains  how  to  subtract  one  number 
from  the  other  so  that  the  remainder  shall  be  seen."* 
Addition  was  performed  just  as  is  done  to-day.  In 
subtraction  for  the  case  of  a  larger  figure  in  the  sub- 
trahend, it  was  the  custom  in  Germany  to  complete 
this  figure  to  10,  to  add  this  complement  to  the  min- 
uend figure,  but  at  the  same  time  to  increase  the  figure 
of  next  higher  order  in  the  subtrahend  by  1  (Fibo- 
nacci's counting-on  method).  In  more  comprehen- 
sive books,  borrowing  for  this  case  was  also  taught. 
Multiplication,  which  presupposed  practice  in  the  mul- 
tiplication table,  was  performed  in  a  variety  of  ways. 
Most  frequently  it  was  effected  as  to-day  with  a  des- 
cent in  steps  by  movement  toward  the  left.  Luca 
Pacioli  describes  eight  different  kinds  of  multiplica- 
tion, among  them  those  above  mentioned,  with  two 
old  Hindu  methods,  one  represented  on  p.  29,  the 
other  cross-multiplication  or  the  lightning  method. 
In  the  latter  method  there  were  grouped  all  the  pro- 
ducts involving  units,  all  those  involving  tens,  all 
those  involving  hundreds.  The  multiplication 

*  Unger,  p.  72. 


46  HISTORY  OF  MATHEMATICS. 

243-139  =  9  -3  + 10(9-4  +  3-3)+  100(9  -2  +  3-4  +  1-3) 
+  1000(2-3  +  1-4)+ 10000-2-1 

was  represented  as  follows : 

4  3 


189 

In  German  books  are  found,  besides  these,  two  note- 
worthy methods  of  multiplication,  of  which  one  be- 
gins on  the  left  (as  with  the  Greeks),  the  partial  pro- 
ducts being  written  in  succession  in  the  proper  place, 
as  shown  by  the  following  example  243  •  839 : 

839 
243 

166867      839  •  243  =  2  •  8  •  1 04  +  2  •  3  •  108  +  2  •  9  •  1 02 
3129  +4-8-108  +  4-3-102  +  4-9-10 

232  +3-8-102  +  3-3-10  +3-9. 

14 

2 
203877 

In  division  the  upwards-division  prevailed ;  it  was 
used  extensively,  although  Luca  Pacioli  in  1494  taught 
the  downwards-division  in  modern  form. 

After  the  completion  of  the  computation,  in  con- 
formity to  historical  tradition,  a  proof  was  demanded. 
At  first  this  was  secured  by  casting  out  the  nines. 
On  account  of  the  untrustworthiness  of  this  method, 
which  Pacioli  perfectly  realised,  the  performance  of 


ARITHMETIC.  47 

the  inverse  operation  was  recommended.  In  course 
of  time  the  use  of  a  proof  was  entirely  given  up. 

Signs  of  operation  properly  so  called  were  not 
yet  in  use ;  in  the  eighteenth  century  they  passed 
from  algebra  into  elementary  arithmetic.  Widmann, 
however,  in  his  arithmetic  has  the  signs  -j-  and  — , 
which  had  probably  been  in  use  some  time  among  the 
merchants,  since  they  appear  also  in  a  Vienna  MS.  of 
the  fifteenth  century.*  At  a  later  time  Wolf  has  the 
sign  -4-  for  minus.  In  numeration  the  first  use  of  the 
word  "million"  in  print  is  due  to  Pacioli  (Summa  de 
Arithmetica,  1494).  Among  the  Italians  the  word  "mil- 
lion" is  said  originally  to  have  represented  a  concrete 
mass,  viz.,  ten  tons  of  gold.  Strangely  enough,  the 
words  "byllion,  tryllion,  quadrillion,  quyllion,  sixlion, 
septyllion,  ottyllion,  nonyllion,"  as  well  as  "million,'' 
are  found  as  early  as  1484  in  Chuquet,  while  the  word 
"miliars"  (equal  to  1000  millions)  is  to  be  traced 
back  to  Jean  Trenchant  of  Lyons  (1588). f 

The  seventeenth  century  was  especially  inventive 
in  instrumental  appliances  for  the  mechanical  per- 
formance of  the  fundamental  processes  of  arithmetic. 
Napier's  rods  sought  to  make  the  learning  of  the  mul- 
tiplication-table superfluous.  These  rods  were  quad- 
rangular prisms  which  bore  on  each  side  the  small 
multiplication-table  for  one  of  the  numbers  1,  2, ...  9. 

*Gerhardt,  Geschichte  der  Mathematik  in  Deutsckland,  1877.  Hereafter 
referred  to  as  Gerhardt. 

tMuller.  Historisch-ftymologitcht  Studien  S6er  mathematiscke  Termino- 
logie.  Hereafter  referred  to  as  Muller. 


48  HISTORY  OF  MATHEMATICS. 

For  extracting  square  and  cube  roots  rods  were  used 
with  the  squares  and  cubes  of  one-figure  numbers  in- 
scribed upon  them.  Real  calculating  machines  which 
gave  results  by  the  simple  turning  of  a  handle,  but  on 
that  account  must  have  proved  elaborate  and  expen- 
sive, were  devised  by  Pascal,  Leibnitz,  and  Matthftus 
Hahn  (1778). 

A  simplification  of  another  kind  was  effected  by 
calculating-tables.  These  were  tables  for  solving 
problems,  accompanied  also  by  very  extended  multi- 
plication-tables, such  as  those  of  Herwart  von  Hohen- 
burg,  from  which  the  product  of  any  two  numbers 
from  1  to  999  could  be  read  immediately. 

For  the  methods  of  computation  of  the  eighteenth 
century  the  arithmetic  writings  of  the  two  Sturms, 
and  of  Wolf  and  Kastner,  are  of  importance.  In  the 
interest  of  commercial  arithmetic  the  endeavor  was 
made  to  abbreviate  multiplication  and  division  by 
various  expedients.  Nothing  essentially  new  was 
gained,  however,  unless  it  be  the  so-called  mental 
arithmetic  or  oral  reckoning  which  in  the  later  decades 
of  this  period  appears  as  an  independent  branch. 

The  nineteenth  century  has  brought  as  a  novelty 
in  elementary  arithmetic  only  the  introduction  of  the 
so-called  Austrian  subtraction  (by  counting  on)  and 
division,  methods  for  which  Fibonacci  had  paved  the 
way.  The  difference  323—187  =  136  is  computed 
by  saying,  7  and  6,  9  and  3,  2  and  1 ;  and  43083  : 185 
is  arranged  as  in  the  first  of  the  following  examples  : 


ARITHMETIC. 


49 


(185 

43083232" 
"608~ 

533 

163 


1 
1 

2 

1679 

2737 

1 
1 

621 

n^r 

1058 

437 

2 

2 

46 

69 

1 

0 

23 

With  sufficient  practice  this  process  certainly  secured 
a  considerable  saving  of  time,  especially  in  the  case 
of  the  determination  of  the  greatest  common  divisor 
of  two  or  more  numbers  as  shown  by  the  second  of 
the  above  examples 

1679  23    73 

2737  ~Tl9* 

2.   Arithmetic  of  Fractions. 

At  the  beginning  of  this  period  reckoning  with 
fractions  was  regarded  as  very  difficult.  The  pupil 
was  first  taught  how  to  read  fractions:  "It  is  to  be 
noticed  that  every  fraction  has  two  figures  with  a  line 
between.  The  upper  is  called  the  numerator,  the 
lower  the  denominator.  The  expression  of  fractions  is 
then:  name  first  the  upper  figure,  then  the  lower,  with 
the  little  word  part  as  f  part"  (Grammateus,  1518).* 
Then  came  rules  for  the  reduction  of  fractions  to  a 
common  denominator,  for  reduction  to  lowest  terms, 
for  multiplication  and  division ;  in  the  last  the  fractions 
were  first  made  to  have  a  common  denominator.  Still 
more  is  found  in  Tartaglia  who  knew  how  to  find  the 
least  common  denominator  ;  in  Stifel  who  performed 


'  Unger,  p.  84. 


50  HISTORY  OF  MATHEMATICS. 

division  by  a  fraction  by  the  use  of  its  reciprocal,  and 
in  the  works  of  other  writers. 

The  way  for  the  introduction  of  decimal  fractions 
was  prepared  by  the  systems  of  sexagesimal  and  duo- 
decimal fractions,  since  by  their  employment  opera- 
tions with  fractions  can  readily  be  performed  by  the 
corresponding  operations  with  whole  numbers.  A  no- 
tation such  as  has  become  usual  in  decimal  fractions 
was  already  known  to  Rudolff,*  who,  in  the  division 
of  integers  by  powers  of  10,  cuts  off  the  requisite 
number  of  places  with  a  comma.  The  complete  knowl- 
edge of  decimal  fractions  originated  with  Simon  Stevin 
who  extended  the  position-system  below  unity  to  any 
extent  desired.  Tenths,  hundredths,  thousandths,  . .  . 
were  called  primes,  sekonties,  terzes  .  .  .;  4.628  is  writ- 
ten 4(0)  6(i)  2(2)  8(3).  Joost  Burgi,  in  his  tables  of  sines, 
perhaps  independently  of  Stevin,  used  decimal  frac- 
tions in  the  form  0.32  and  3.2.  The  introduction  of 
the  comma  as  a  decimal  point  is  to  be  assigned  to 
Kepler. f  In  practical  arithmetic,  aside  from  logarith- 
mic computations,  decimal  fractions  were  used  only 
in  computing  interest  and  in  reduction-tables.  They 
were  brought  into  ordinary  arithmetic  at  the  begin- 
ning of  the  nineteenth  century  in  connection  with  the 
introduction  of  systems  of  decimal  standards. 

*  Gerhardt. 

tThe  first  use  of  the  decimal  point  is  found  in  the  trigonometric  tables 
of  Pitiscus,  1612.  Cantor,  II.,  p.  555. 


ARITHMETIC.  51 

3.   Applied  Arithmetic. 

During  the  transition  period  of  the  Middle  Ages 
applied  arithmetic  had  absorbed  much  from  the  Latin 
treatises  in  a  superficial  and  incomplete  manner ;  the 
fifteenth  and  sixteenth  centuries  show  evidences  of 
progress  in  this  direction  also.  Even  the  Bamberger 
Arithmetic  of  1483  bears  an  exclusively  practical  stamp 
and  aims  only  at  facility  of  computation  in  mercan- 
tile affairs.  That  method  of  solution  which  in  the 
books  on  arithmetic  everywhere  occupied  the  first 
place  was  the  "regeldetri"  (regula  de  tri,  rule  of 
three),  known  also  as  the  "merchant's  rule,"  or 
"golden  rule."*  The  statement  of  the  rule  of  three 
was  purely  mechanical ;  so  little  thought  was  bestowed 
upon  the  accompanying  proportion  that  even  master 
accountants  were  content  to  write  4  fl  12  ft  20  fl?  in- 
stead of  4  fl  :  20  fl  =  12  ft  :  x  ft.f  There  can  indeed 
be  found  examples  of  the  rule  of  three  with  indirect 
ratios,  but  with  no  explanations  of  any  kind  whatever. 
Problems  involving  the  compound  rule  of  three  (regula 
de  quinque,  etc. )  were  solved  merely  by  successive  ap- 
plications of  the  simple  rule  of  three.  In  Tartaglia 
and  Widmann  we  find  equation  of  payments  treated 
according  to  the  method  still  in  use  to-day.  Other- 
wise, Widmann's  Arithmetic  of  1489  shows  great  ob- 
scurity and  lack  of  scope  in  rules  and  nomenclature, 
so  that  not  infrequently  the  same  matter  appears  un- 

*  Cantor,  II.,  p.  205  :  Unger,  p.  86.        t  Cantor,  II.,  p.  368;  Unger,  p.  87. 


52  HISTORY  OF  MATHEMATICS. 

der  different  names.  He  introduces  "Regula  Residui, 
Reciprocationis,  Excessus,  Divisionis,  Quadrata,  In- 
ventionis,  Fusti,  Transversa,  Ligar,  Equalitatis,  Legis, 
Augmenti,  Augmenti  et  Decrementi,  Sententiarum, 
Suppositionis,  Collectionis,  Cubica,  Lucri,  Pagamenti, 
Alligationis,  Falsi,"  so  that  in  later  years  Stifel  did  not 
hesitate  to  declare  these  things  simply  laughable.* 
Problems  of  proportional  parts  and  alligation  were 
solved  by  the  use  of  as  many  proportions  as  corre- 
sponded to  the  number  of  groups  to  be  separated. 
For  the  computation  of  compound  interest  Tartaglia 
gave  four  methods,  among  them  computation  by  steps 
from  year  to  year,  or  computation  with  the  aid  of 
the  formula  b  =  aqn,  although  he  does  not  give  this 
formula.  Computing  of  exchange  was  taught  in  its 
most  simple  form.  It  is  said  that  bills  of  exchange 
were  first  used  by  the  Jews  who  migrated  into  Lom- 
bardy  after  being  driven  from  France  in  the  seventh 
century.  The  Ghibellines  who  fled  from  Lombardy 
introduced  exchange  into  Amsterdam,  and  from  this 
city  its  use  spread. f  In  1445  letters  of  exchange  were 
brought  to  Nuremberg. 

The  chain  rule  {Kettensatz},  essentially  an  Indian 
method  which  is  described  by  Brahmagupta,  was  de- 
veloped during  the  sixteenth  century,  but  did  not 
come  into  common  use  until  two  centuries  later.  The 
methods  of  notation  differed.  Pacioli  and  Tartaglia 

•Treutlein,  Die  deutsche  Coss,  Schlomilch's  Zeitschrift,  Bd.  24,  HI.  A. 
t  Unger,  p.  90. 


ARITHMETIC.  53 

wrote  all  numbers  in  a  horizontal  line  and  multiplied 
terms  of  even  and  of  odd  order  into  separate  products. 
Stifel  proceeded  in  the  same  manner,  only  he  placed 
all  terms  vertically  beneath  one  another.  In  the 
work  of  Rudolff,  who  also  saw  the  advantage  of  can- 
cellation, we  find  the  modern  method  of  representing 
the  chain  rule,  but  the  answer  comes  at  the  end.* 

About  this  time  a  new  method  of  reckoning  was 
introduced  from  Italy  into  Germany  by  the  merchants, 
which  came  to  occupy  an  important  place  in  the  six- 
teenth century,  and  still  more  so  in  the  seventeenth. 
This  Welsh  (i.  e.,  foreign)  practice,  as  it  soon  came 
to  be  called,  found  its  application  in  the  development 
of  the  product  of  two  terms  of  a  proportion,  especially 
when  these  were  unlike  quantities.  The  multiplier, 
together  with  the  fraction  belonging  to  it,  was  sepa- 
rated into  its  addends,  to  be  derived  successively  one 
from  another  in  the  simplest  possible  manner.  How 
well  Stifel  understood  the  real  significance  and  appli- 
cability of  the  Welsh  practice,  the  following  statement 
shows  :f  "The  Welsh  practice  is  nothing  more  than 
a  clever  and  entertaining  discovery  in  the  rule  of  three. 
But  let  him  who  is  not  acquainted  with  the  Welsh 
practice  rely  upon  the  simple  rule  of  three,  and  he 
will  arrive  at  the  same  result  which  another  obtains 
through  the  Welsh  practice."  At  this  time,  too,  we 
find  tables  of  prices  and  tables  of  interest  in  use, 
their  introduction  being  also  ascribable  to  the  Italians. 

*  Unger,  p.  92.  1  Unger,  p.  94. 


54  HISTORY  OF  MATHEMATICS. 

In  the  sixteenth  century  we  also  come  upon  examples 
for  the  regula  virginum  and  the  regula  falsi  in  writings 
intended  for  elementary  instruction  in  arithmetic, — 
writings  into  which,  ordinarily,  was  introduced  all  the 
learning  of  the  author.  The  significance  of  these 
rules,  however,  does  not  lie  in  the  realm  of  elemen- 
tary arithmetic,  but  in  that  of  equations.  In  the 
same  way,  a  few  arithmetic  writings  contained  direc- 
tions for  the  construction  of  magic  squares,  and  most 
of  them  also  contained,  as  a  side-issue,  certain  arith- 
metic puzzles  and  humorous  questions  (Rudolff  calls 
them  Schimpfrechnung).  The  latter  are  often  mere 
disguises  of  algebraic  equations  (the  problem  of  the 
hound  and  the  hares,  of  the  keg  with  three  taps,  of 
obtaining  a  number  which  has  been  changed  by  cer- 
tain operations,  etc.). 

The  seventeenth  century  brought  essential  innova- 
tions only  in  the  province  of  commercial  computation. 
While  the  sixteenth  century  was  in  possession  of  cor- 
rect methods  in  all  computations  of  interest  when 
the  amount  at  the  end  of  a  given  time  was  sought, 
there  were  usually  grave  blunders  when  the  principal 
was  to  be  obtained,  that  is,  in  computing  the  discount 
on  a  given  sum.  The  discount  in  100  was  computed 
somewhat  in  this  manner:*  100  dollars  gives  after 
two  years  10  dollars  in  interest ;  if  one  is  to  pay  the 
100  dollars  immediately,  deduct  10  dollars."  No  less 
a  man  than  Leibnitz  pointed  out  that  the  discount 

*Unger,  p.  132. 


ARITHMETIC.  55 

must  be  reckoned  upon  100.  Among  the  majority  of 
arithmeticians  his  method  met  with  the  misunder- 
standing that  if  the  discount  at  5%  for  one  year  is  ^ 
the  discount  for  two  years  must  be  72T.  It  was  not 
until  the  eighteenth  century,  after  long  and  sharp 
controversy,  that  mathematicians  and  jurists  united 
upon  the  correct  formula. 

In  the  computation  of  exchange  the  Dutch  were 
essentially  in  advance  of  other  peoples.  They  pos- 
sessed special  treatises  in  this  line  of  commercial  arith- 
metic and  through  them  they  were  well  acquainted 
with  the  fundamental  principles  of  the  arbitration  of 
exchange.  In  the  way  of  commercial  arithmetic  many 
expedients  were  discovered  in  the  eighteenth  century 
to  aid  in  the  performance  of  the  fundamental  opera- 
tions and  in  solving  concrete  problems.  Calculation 
of  exchange  and  arbitration  of  exchange  were  firmly 
established  and  thoroughly  discussed  by  Clausberg. 
Especial  consideration  was  given  to  what  was  called 
the  Reesic  rule,  which  was  looked  upon  as  differing 
from  the  well-known  chain-rule.  Rees's  book,  which 
was  written  in  Dutch,  was  translated  into  French  in 
1737,  and  from  this  language  into  German  in  1739. 
In  the  construction  of  his  series  Rees  began  with  the 
required  term ;  in  the  computation  the  elimination  of 
fractions  and  cancellation  came  first,  and  then  fol- 
lowed the  remaining  operations,  multiplication  and 
division. 

Computation  of  capital  and  interest  was  extended, 


56  HISTORY  OF  MATHEMATICS. 

through  the  establishment  of  insurance  associations, 
to  a  so-called  political  arithmetic,  in  which  calcula- 
tion of  contingencies  and  annuities  held  an  important 
place. 

The  first  traces  of  conditions  for  the  evolution  of 
a  political  arithmetic*  date  back  to  the  Roman  prefect 
Ulpian,  who  about  the  opening  of  the  third  century 
A.D.  projected  a  mortality  table  for  Roman  subjects,  f 
But  there  are  no  traces  among  the  Romans  of  life  in- 
surance institutions  proper.  It  is  not  until  the  Middle 
Ages  that  a  few  traces  appear  in  the  legal  regulations 
of  endowments  and  guild  finances.  From  the  four- 
teenth century  there  existed  travel  and  accident  in- 
surance companies  which  bound  themselves,  in  con- 
sideration of  the  payment  of  a  certain  sum,  to  ransom 
the  insured  from  captivity  among  the  Turks  or  Moors. 

Among  the  guilds  of  the  Middle  Ages  the  idea  of 
association  for  mutual  assistance  in  fires,  loss  of  cattle 
and  similar  losses  had  already  assumed  definite  shape. 
To  a  still  more  marked  degree  was  this  the  case  among 
the  guilds  of  artisans  which  arose  after  the  Reforma- 
tion— guilds  which  established  regular  sick  and  burial 
funds. 

We  must  consider  tontines  as  the  forerunner  of 
annuity  insurance.  In  the  middle  of  the  seventeenth 
century  an  Italian  physician,  Lorenzo  Tonti,  induced 
a  number  of  persons  in  Paris  to  contribute  sums  of 

*Karup,  Theoretisches  Handbuch  der  Ltbenrversicherung.     1871. 
t  Cantor,  I.,  p.  522. 


ARITHMETIC.  57 

money  the  interest  of  which  should  be  divided  annu- 
ally among  the  surviving  members.  The  French  gov- 
ernment regarded  this  procedure  as  an  easy  method 
of  obtaining  money  and  established  from  1689  to  1759 
ten  state  tontines  which,  however,  were  all  given  up 
in  1770,  as  it  had  been  proved  that  this  kind  of  state 
loan  was  not  lucrative. 

In  the  meantime  two  steps  had  been  taken  which, 
by  using  the  results  of  mathematical  science,  provided 
a  secure  foundation  for  the  business  of  insurance. 
Pascal  and  Fermat  had  outlined  the  calculation  of 
contingencies,  and  the  Dutch  statesman  De  Witt  had 
made  use  of  their  methods  to  lay  down  in  a  separate 
treatise  the  principles  of  annuity  insurance  based  upon 
the  birth  and  death  lists  of  several  cities  of  Holland. 
On  the  other  hand,  Sir  William  Petty,  in  1662,  in  a 
work  on  political  arithmetic*  contributed  the  first  val- 
uable investigations  concerning  general  mortality — a 
work  which  induced  John  Graunt  to  construct  mor- 
tality tables.  Mortality  tables  were  also  published  by 
Kaspar  Neumann,  a  Breslau  clergyman,  in  1692,  and 
these  attracted  such  attention  that  the  Royal  Society 
of  London  commissioned  the  astronomer  Halley  to 
verify  these  tables.  With  the  aid  of  Neumann's  ma- 
terial Halley  constructed  the  first  complete  tables  of 
mortality  for  the  various  ages.  Although  these  tables 
did  not  obtain  the  recognition  they  merited  until  half 
a  century  later,  they  furnished  the  foundation  for  all 

*  Recently  republished  in  inexpensive  form  in  Cassell's  National  Library. 


58  HISTORY  OF  MATHEMATICS. 

later  works  of  this  kind,  and  hence  Halley  is  justly 
called  the  inventor  of  mortality  tables. 

The  first  modern  life-insurance  institutions  were 
products  of  English  enterprise.  In  the  years  1698  and 
1699  there  arose  two  unimportant  companies  whose 
field  of  operations  remained  limited.  In  the  year 
1705,  however,  there  appeared  in  London  the  "Amic- 
able" which  continued  its  corporate  existence  until 
1866.  The  "Royal  Exchange"  and  "London  Assur- 
ance Corporation,"  two  older  associations  for  fire  and 
marine  insurance,  included  life  insurance  in  their  busi- 
ness in  1721,  and  are  still  in  existence.  There  was  soon 
felt  among  the  managers  of  such  institutions  the  im- 
perative need  for  reliable  mortality  tables,  a  fact  which 
resulted  in  Halley's  work  being  rescued  from  oblivion 
by  Thomas  Simpson,  and  in  James  Dodson's  project- 
ing the  first  table  of  premiums,  on  a  rising  scale,  after 
Halley's  method.  The  oldest  company  which  used 
as  a  basis  these  scientific  innovations  was  the  "  Society 
for  Equitable  Assurances  on  Lives  and  Survivorships," 
founded  in  1765. 

While  at  the  beginning  of  the  nineteenth  century 
eight  life  insurance  companies  were  already  carrying 
on  their  beneficent  work  in  England,  there  was  at  the 
same  time  not  a  single  institution  of  this  kind  upon 
the  Continent,  in  spite  of  the  progress  which  had  been 
made  in  the  science  of  insurance  by  Leibnitz,  the  Ber- 
noullis,  Euler  and  others.  In  France  there  appeared  in 
1819  "La  compagnie  d'  assurances  generates  sur  la 


ARITHMETIC.  59 

vie."  In  Bremen  the  founding  of  a  life  insurance  com- 
pany was  frustrated  by  the  disturbances  of  the  war  in 
1806.  It  was  not  until  1828  that  the  two  oldest  Ger- 
man companies  were  formed,  the  one  in  Lubeck,  the 
other  in  Gotha  under  the  management  of  Ernst  Wil- 
helm  Arnoldi,  the  "Father  of  German  Insurance." 

The  nineteenth  century  has  substantially  enriched 
the  literature  of  mortality  tables,  in  such  tables  as 
those  compiled  by  the  Englishmen  Arthur  Morgan 
(in  the  eighteenth  century)  and  Farr,  by  the  Belgian 
Quetelet,  and  by  the  Germans,  Brune,  Heym,  Fischer, 
Wittstein,  and  Scheffler.  A  recent  acquisition  in  this 
field  is  the  table  of  deaths  compiled  in  accordance 
with  the  vote  of  the  international  statistical  congress 
at  Budapest  in  1876,  which  gives  the  mortality  of  the 
population  of  the  German  Empire  for  the  ten  years 
1871-1881.  Further  development  and  advancement 
of  the  science  of  insurance  is  provided  for  by  the 
"  Institute  of  Actuaries  "  founded  in  London  in  1849 — 
an  academic  school  with  examinations  in  all  branches 
of  the  subject.  There  has  also  been  in  Berlin  since 
1868  a  "College  of  the  Science  of  Insurance,"  but 
it  offers  no  opportunity  for  study  and  no  examina- 
tions. 

The  following  compilations  furnish  a  survey  of  the 
conditions  of  insurance  in  the  year  1890  and  of  its 
development  in  Germany.*  There  were  in  Germany: 

*Karnp,  Theoretisches  Handbuch  der  Lebensver sicker  ung,  1871.  Johnson, 
Universal  Cyclopedia,  under  "  Life-insurance." 


6o 


HISTORV  OF  MATHEMATICS. 


NUMBER  OF 
LIFE  INS. 

co's. 

NUMBER  OF 
PERSONS 
INSURED 

FOR  THE  SUM 

IN  ROUND  NUMBER! 
(MILLION  MARKS) 

12 

46,980 

170 

20 

90,128 

300 

32 
49 

305,433 

900 
4250 

1852 
1858 
1866 
1890 


There  were  in  1890  : 


INSURANCE  CO'S. 

Germany 49 

Great  Britain  and  Ireland 75 

France 17 

Rest  of  Europe 58 

United  States  of  America 48 


NUMBER  OF  LIFE          AMOUNT  OF  INSURANCE 
IN  FORCE 

4250  million  marks 
900  ' '  pounds 
3250  "  francs 
3200  "  francs 
4000  "  dollars 


All  that  the  eighteenth  century  developed  or  dis- 
covered has  been  further  advanced  in  the  nineteenth. 
The  center  of  gravity  of  practical  calculation  lies  in 
commercial  arithmetic.  This  is  also  finding  expres- 
sion in  an  exceedingly  rich  literature  which  has  been 
extended  in  an  exhaustive  manner  in  all  its  details, 
but  which  contains  nothing  essentially  new  except  the 
methods  of  calculating  interest  in  accounts  current. 


III.  ALGEBRA. 

A.   GENERAL  SURVEY. 

THE  beginnings  of  general  mathematical  science 
are  the  first  important  outcome  of  special  studies 
of  number  and  magnitude ;  they  can  be  traced  back 
to  the  earliest  times,  and  their  circle  has  only  gradu- 
ally been  expanded  and  completed.  The  first  period 
reaches  up  to  and  includes  the  learning  of  the  Arabs ; 
its  contributions  culminate  in  the  complete  solution 
of  the  quadratic  equation  of  one  unknown  quantity, 
and  in  the  trial  method,  chiefly  by  means  of  geometry, 
of  solving  equations  of  the  third  and  fourth  degrees. 
The  second  period  includes  the  beginning  of  the 
development  of  the  mathematical  sciences  among  the 
peoples  of  the  West  from  the  eighth  century  to  the 
middle  of  the  seventeenth.  The  time  of  Gerbert  forms 
the  beginning  and  the  time  of  Kepler  the  end  of  this 
period.  Calculations  with  abstract  quantities  receive 
a  material  simplification  in  form  through  the  use  of 
abbreviated  expressions  for  the  development  of  for- 
mulae ;  the  most  important  achievement  lies  in  the 
purely  algebraic  solution  of  equations  of  the  third  and 
fourth  degrees  by  means  of  radicals. 


62  HISTORY  OF  MATHEMATICS. 

The  third  period  begins  with  Leibnitz  and  Newton 
and  extends  from  the  middle  of  the  seventeenth  cen- 
tury to  the  present  time.  In  the  first  and  larger  part 
of  this  period  a  new  light  was  diffused  over  fields 
which  up  to  that  time  had  been  only  partially  ex- 
plored, by  the  discovery  of  the  methods  of  higher 
analysis.  At  the  end  of  this  first  epoch  there  appeared 
certain  mathematicians  who  devoted  themselves  to 
the  study  of  combinations  but  failed  to  reach  the 
lofty  points  of  view  of  a  Leibnitz.  Euler  and  La- 
grange,  thereupon,  assumed  the  leadership  in  the  field 
of  pure  analysis.  Euler  led  the  way  with  more  than 
seven  hundred  dissertations  treating  all  branches  of 
mathematics.  The  name  of  the  great  Gauss,  who 
drew  from  the  works  of  Newton  and  Euler  the  first 
nourishment  for  his  creative  genius,  adorns  the  be- 
ginning of  the  second  epoch  of  the  third  period. 
Through  the  publication  of  more  than  fifty  large 
memoirs  and  a  number  of  smaller  ones,  not  alone  on 
mathematical  subjects  but  also  on  physics  and  astron- 
omy, he  set  in  motion  a  multitude  of  impulses  in  the 
most  varied  directions.  At  this  time,  too,  there  opened 
new  fields  in  which  men  like  Abel,  Jacobi,  Cauchy, 
Dirichlet,  Riemann,  Weierstrass  and  others  have  made 
a  series  of  most  beautiful  discoveries. 


B.    FIRST  PERIOD. 

FROM  THE  EARLIEST  TIMES  TO  THE  ARABS. 

I.    General  Arithmetic. 

However  meagre  the  information  which  describes 
the  evolution  of  mathematical  knowledge  among  the 
earliest  peoples,  still  we  find  isolated  attempts  among 
the  Egyptians  to  express  the  fundamental  processes 
by  means  of  signs.  In  the  earliest  mathematical  pa- 
pyrus *  we  find  as  the  sign  of  addition  a  pair  of  walk- 
ing legs  travelling  in  the  direction  toward  which  the 
birds  pictured  are  looking.  The  sign  for  subtraction 
consists  of  three  parallel  horizontal  arrows.  The  sign 
for  equality  is  <^.  Computations  are  also  to  be  found 
which  show  that  the  Egyptians  were  able  to  solve  sim- 
ple problems  in  the  field  of  arithmetic  and  geometric 
progressions.  The  last  remark  is  true  also  of  the 
Babylonians.  They  assumed  that  during  the  first  five 
of  the  fifteen  days  between  new  moon  and  full  moon, 
the  gain  in  the  lighted  portion  of  its  disc  (which  was 
divided  into  240  parts)  could  be  represented  by  a  geo 
metric  progression,  during  the  ten  following  days  by 
an  arithmetic  progression.  Of  the  240  parts  there 
were  visible  on  the  first,  second,  third  .  .  .  fifteenth 
day 

*  Cantor,  I.,  p.  37. 


64  HISTORY  OF  MATHEMATICS. 

5  10  20  40  1.20 
1.36  1.52  2.08  2.24  2.40 
2.56  3.12  3.28  3.44  4. 

The  system  of  notation  is  sexagesimal,  so  that  we  are 
to  take  3.28  =  3x60  +  28  =  208.*  Besides  this  there 
have  been  found  on  ancient  Babylonian  monuments 
the  first  sixty  squares  and  the  first  thirty-two  cubes 
in  the  sexagesimal  system  of  notation. 

The  spoils  of  Greek  treasures  are  far  richer.  Even 
the  name  of  the  entire  science  y  [jia6r)tM.TiKTJ  comes  from 
the  Greek  language.  In  the  time  of  Plato  the  word 
(ia.Oijfw.Ta.  included  all  that  was  considered  worthy  of 
scientific  instruction.  It  was  not  until  the  time  of  the 
Peripatetics,  when  the  art  of  computation  (logistic^ 
and  arithmetic,  plane  and  solid  geometry,  astronomy 
and  music  were  enumerated  in  the  list  of  mathemat- 
ical sciences,  that  the  word  received  its  special  signifi- 
cance. Especially  with  Heron  of  Alexandria  logistic 
appears  as  elementary  arithmetic,  while  arithmetic  so 
called  is  a  science  involving  the  theory  of  numbers. 

Greek  arithmetic  and  algebra  appeared  almost 
always  under  the  guise  of  geometry,  although  the 
purely  arithmetic  and  algebraic  method  of  thinking 
was  not  altogether  lacking,  especially  in  later  times. 
Aristotlef  is  familiar  with  the  representation  of  quan- 
tities by  letters  of  the  alphabet,  even  when  those 
quantities  do  not  represent  line-segments  ;  he  says  in 

•Cantor,  I.,  p.  81.  t Cantor,  I.,  p.  240. 


ALGEBRA.  65 

one  place  :  "  If  A  is  the  moving  force,  B  that  which  is 
moved,  F  the  distance,  and  A  the  time,  etc."  By  the 
time  of  Pappus  there  had  already  been  developed  a 
kind  of  reckoning  with  capital  letters,  since  he  was 
able  to  distinguish  as  many  general  quantities  as  there 
were  such  letters  in  the  alphabet.  (The  small  letters 
a,  ft,  y,  stood  for  the  numbers  1,  2,  3,  .  .  .)  Aristotle 
has  a  special  word  for  "continuous"  and  a  definition 
for  continuous  quantities.  Diophantus  went  farther 
than  any  of  the  other  Greek  writers.  With  him  there 
already  appear  expressions  for  known  and  unknown 
quantities.  Hippocrates  calls  the  square  of  a  number 
Swa/us  (power),  a  word  which  was  transferred  to  the 
Latin  as  potentia  and  obtained  later  its  special  mathe- 
matical significance.  Diophantus  gives  particular 
names  to  all  powers  of  unknown  quantities  up  to  the 
sixth,  and  introduces  them  in  abbreviated  forms,  so 
that  x2,  x3,  x*,  x6,  x*,  appear  as  8s,  K°,  8S°,  SK°,  KKS. 
The  sign  for  known  numbers  is  p?.  In  subtraction 
Diophantus  makes  use  of  the  sign  /p  (an  inverted  and 
abridged  if/) ;  i,  an  abbreviation  for  Zo-oi,  equal,  appears 
as  the  sign  of  equality.  A  term  of  an  expression  is 
called  etSos;  this  word  went  into  Latin  as  species  and 
was  used  in  forming  the  title  arithmetica  speciosa=.3\- 
gebra.*  The  formulae  are  usually  given  in  words  and 
are  represented  geometrically,  as  long  as  they  have  to 
do  only  with  expressions  of  the  second  dimension.  The 
first  ten  propositions  in  the  second  book  of  Euclid, 

*  Cantor,  I.,  p.  442. 


66  HISTORY  OF  MATHEMATICS. 

for  example,  are  enunciations  in  words  and  geometric 
figures,  and  correspond  among  others  to  the  expres- 
sions a(b+c  +  d.  .  ,}—ab  +  ac -\-ad-\- ,  (a -f£)2 

=  a2  -f  2a&  -f  t>*  =(a  -f  £)  a  +  (a  +  V)b. 

Geometry  was  with  the  Greeks  also  a  means  for  in- 
vestigations in  the  theory  of  numbers.  This  is  seen, 
for  instance,  in  the  remarks  concerning  gnomon-num- 
bers. Among  the  Pythagoreans  a  square  out  of  which 
a  corner  was  cut  in  the  shape  of  a  square  was  called  a 
gnomon.  Euclid  also  used  this  expression  for  the 
figure  ABCDEF  which  is  obtained  from  the  parallelo- 
gram ABCB'  by  cutting  out  the  parallelogram  DB'FE. 
The  gnomon-number  of  the  Pythagoreans  is  20+  1  ; 
for  when  ABCB'  is  a  square,  the  square  upon  DE  =  n 


can  be  made  equal  to  the  square  on£C=n-\-l  by 
adding  the  square  BE=l  X  1  and  the  rectangles  AE 
=  C£  =  lXn,  since  we  have  «2-j-  2»-j-  1  =  (»+  I)2. 
Expressions  like  plane  and  solid  numbers  used  for 
the  contents  of  spatial  magnitudes  of  two  and  three 
dimensions  also  serve  to  indicate  the  constant  tend- 


ALGEBRA.  67 

ency  to  objectify  mathematical  thought  by  means  of 
geometry. 

All  that  was  known  concerning  numbers  up  to  the 
third  century  B.  C. ,  Euclid  comprehended  in  a  general 
survey.  In  his  Elements  he  speaks  of  magnitudes,  with- 
out, however,  explaining  this  concept,  and  he  under- 
stands by  this  term,  besides  lines,  angles,  surfaces 
and  solids,  the  natural  numbers.*  The  difference  be- 
tween even  and  odd,  between  prime  and  composite 
numbers,  the  method  for  finding  the  least  common 
multiple  and  the  greatest  common  divisor,  the  con- 
struction of  rational  right  angled  triangles  according 
to  Plato  and  the  Pythagoreans — all  these  are  familiar 
to  him.  A  method  (the  "sieve")  for  sorting  out 
prime  numbers  originated  with  Eratosthenes.  It  con- 
sists in  writing  down  all  the  odd  numbers  from  3 
on,  and  then  striking  out  all  multiples  of  3,  5,  7  ... 
Diophantus  stated  that  numbers  of  the  form  a2  -)-  2ab 
-f-  &*  represent  a  square  and  also  that  numbers  of  the 
form  (a2  -f-  <£2)  (V2  -|-  </2)  can  represent  a  sum  of  two 
squares  in  two  ways  ;  for  (ac  -\-  bd}*  -J-  (ad —  bcf  = 
(ac  —  bd^  +  (ad+  bcf  =  (a2  -f  £2)  (V2  -f  </2). 

The  knowledge  of  the  Greeks  in  the  field  of  ele- 
mentary series  was  quite  comprehensive.  The  Pythag- 
oreans began  with  the  series  of  even  and  odd  num- 
bers. The  sum  of  the  natural  numbers  gives  the 
triangular  number,  the  sum  of  the  odd  numbers  the 
square,  the  sum  of  the  even  numbers  gives  the  hetero- 

*Treutlein. 


68  HISTORY  OF  MATHEMATICS. 

mecic  (oblong)  number  of  the  form  n(n-\- 1).  Square 
numbers  they  also  recognised  as  the  sum  of  two  suc- 
cessive triangular  numbers.  The  Neo-Pythagoreans 
and  the  Neo-Platonists  made  a  study  not  only  of  po- 
lygonal but  also  of  pyramidal  numbers.  Euclid  treated 
geometrical  progressions  in  his  Elements.  He  ob- 
tained the  sum  of  the  series  l  +  2-(-4-|-8.  .  .  and 
noticed  that  when  the  sum  of  this  series  is  a  prime 
number,  a  "perfect  number"  results  from  multiply- 
ing it  by  the  last  term  of  the  series  (l-j-2-j-  4  =  7; 
7X4  =  28;  28  =  1  +  2  +  4+  7+  14;  cf.  p.  35).  In- 
finite convergent  series  appear  frequently  in  the  works 
of  Archimedes  in  the  form  of  geometric  series  whose 
ratios  are  proper  fractions ;  for  example,  in  calculating 
the  area  of  the  segment  of  a  parabola,  where  the  value 
of  the  series  1  +  J  +  iV  +  •  •  •  is  found  to  be  £.  He 
also  performs  a  number  of  calculations  for  obtaining 
the  sum  of  an  infinite  series  for  the  purpose  of  esti- 
mating areas  and  volumes.  His  methods  are  a  sub- 
stitute for  the  modern  methods  of  integration,  which 
are  used  in  cases  of  this  kind,  so  that  expressions  like 

rx  dx = $£*,  Cx*  dx = ^3 

0  0 

and  other  similar  expressions  are  in  their  import  and 
essence  quite  familiar  to  him.* 

The  introduction  of  the  irrational  is  to  be  traced 
back  to  Pythagoras,  since  he  recognised  that  the  hy- 

*Zeuthen,  Die  Lehre  von  den  KegeUchnitten  im  Altertttm.     Deutsch  von 
v.  Fischer-Benzon.    1886. 


69 


potenuse  of  a  right-angled  isosceles  triangle  is  in- 
commensurable with  its  sides.  The  Pythagorean 
Theodorus  of  Cyrene  proved  the  irrationality  of  the 
square  roots  of  3,  5,  7,  ...  17.* 

Archytas  classified  numbers  in  general  as  rational 
and  irrational.  Euclid  devoted  to  irrational  quantities 
a  particularly  exhaustive  investigation  in  his  Ele- 
ments, a  work  which  belongs  to  the  domain  of  Arith- 
metic as  much  as  to  that  of  Geometry.  Three  books 
among  the  thirteen,  the  seventh,  eighth  and  ninth, 
are  of  purely  arithmetic  contents,  and  in  the  tenth 
book  there  appears  a  carefully  wrought-out  theory  of 
"Incommensurable  Quantities,"  that  is,  of  irrational 
quantities,  as  well  as  a  consideration  of  geometric 
ratios.  At  the  end  of  this  book  Euclid  shows  in  a 
very  ingenious  manner  that  the  side  of  a  square  and 
its  diagonal  are  incommensurable ;  the  demonstration 
culminates  in  the  assertion  that  in  the  case  of  a  ra- 
tional relationship  between  these  two  quantities  a 
number  must  have  at  the  same  time  the  properties  of 
an  even  and  an  odd  number,  f  In  his  measurement 
of  the  circle  Archimedes  calculated  quite  a  number  of 
approximate  values  for  square  roots ;  for  example, 

1351         /!r       265 

T8ir>1/3>T53- 
Nothing  definite,  however,  is  known  concerning  the 

*  Cantor,  I.,  p.  170. 

t  Montucla,  I.,  p.  208.  Montucla  says  that  he  knew  an  architect  who  lived 
in  the  firm  conviction  that  the  square  root  of  2  could  be  represented  as  a 
ratio  of  finite  integers,  and  who  assured  him  that  by  this  method  he  had 
already  reached  the  looth  decimal. 


70  HISTORY  OF  MATHEMATICS. 

method  he  used.  Heron  also  was  acquainted  with 
such  approximate  values  (£  instead  of  1/2,  f|  instead 
of  1/3);*  and  although  he  did  not  shrink  from  the 
labor  of  obtaining  approximate  values  for  square 
roots,  in  the  majority  of  cases  he  contented  himself 
with  the  well-known  approximation  V/at-^b  =  a±-^-, 
e.  g.,  1/63  =  I/ 82  — 1=8  —  ^.  Incase  greater  ex- 
actness was  necessary,  Heron  f  used  the  formula 
l/««+T=  0  +  T  +  y  +  T+---  Incidentally  he  used 
the  identity  ~\/a2t>  =  al/J  and  asserted,  for  example, 
that  1/108"=  1/6^3  =  6i/3  =  6-  $£  =  10  +  £  +  ^. 
Moreover,  we  find  in  Heron's  Stereometrica  the  first 
example  of  the  square  root  of  a  negative  number, 
namely  1/81  — 144,  which,  however,  without  further 
consideration,  is  put  down  by  the  computer  as  8  less 
•jJ^,  which  shows  that  negative  quantities  were  un- 
known among  the  Greeks.  It  is  true  that  Diophantus 
employed  differences,  but  only  those  in  which  the 
minuend  was  greater  than  the  subtrahend.  Through 
Theon  we  are  made  acquainted  with  another  method 
of  extracting  the  square  root;  it  corresponds  with  the 
method  in  use  at  present,  with  the  exception  that  the 
Babylonian  sexagesimal  fractions  are  used,  as  was 
customary  until  the  introduction  of  decimal  fractions. 
Furthermore,  we  find  in  Aristotle  traces  of  the 
theory  of  combinations,  and  in  Archimedes  an  at- 
tempt at  the  representation  of  a  quantity  which  in- 

*  Cantor,  I.,  p.  368.  t  Tannery  in  Bordeaux  Mtm.,  IV.,  1881. 


ALGEBRA.  JI 

creases  beyond  all  limits,  first  in  his  extension  of  the 
number-system,  and  then  in  his  work  entitled  \l/afi- 
lump  (Latin  arenarius,  the  sand-reckoner).  Archi- 
medes arranges  the  first  eight  orders  of  the  decimal 
system  together  in  an  octad ;  108  octads  constitute  a 
period,  and  then  these  periods  are  arranged  again 
according  to  the  same  law.  In  the  sand-reckoning, 
Archimedes  solves  the  problem  of  estimating  the 
number  of  grains  of  sand  that  can  be  contained  in  a 
sphere  which  includes  the  whole  universe.  He  as- 
sumes that  10,000  grains  of  sand  take  up  the  space  of 
a  poppy-seed,  and  he  finds  the  sum  of  all  the  grains 
to  be  10  000  000  units  of  the  eighth  period  of  his  sys- 
tem, or  1063.  It  is  possible  that  Archimedes  in  these 
observations  intended  to  create  a  counterpart  to  the 
domain  of  infinitesimal  quantities  which  appeared  in 
his  summations  of  series,  a  counterpart  not  accessible 
to  the  ordinary  arithmetic. 

In  the  fragments  with  which  we  are  acquainted 
from  the  writings  of  Roman  surveyors  (agrimensores) 
there  are  but  few  arithmetic  portions,  these  having 
to  do  with  polygonal  and  pyramidal  numbers.  Ob- 
viously they  are  of  Greek  origin,  and  the  faulty  style 
in  parts  proves  that  there  was  among  the  Romans  no 
adequate  comprehension  of  matters  of  this  kind. 

The  writings  of  the  Hindu  mathematicians  are  ex- 
ceedingly rich  in  matters  of  arithmetic.  Their  sym- 
bolism was  quite  highly  developed  at  an  early  date.* 

*  Cantor,  I.,  p.  558. 


72  HISTORY  OF  MATHEMATICS. 

Aryabhatta  calls  the  unknown  quantity  gulika  ("little 
ball"),  later  yavattavat,  or  abbreviated 70  ("as  much 
as").  The  known  quantity  is  called  rupaka  or  ru 
("coin").  If  one  quantity  is  to  be  added  to  another, 
it  is  placed  after  it  without  any  particular  sign.  The 
same  method  is  followed  in  subtraction,  only  in  this 
case  a  dot  is  placed  over  the  coefficient  of  the  subtra- 
hend so  that  positive  (dhana,  assets)  and  negative  quan- 
tities (kshaya,  liabilities)  can  be  distinguished.  The 
powers  of  a  quantity  also  receive  special  designations. 
The  second  power  is  varga  or  va,  the  third  ghana  or 
gha,  the  fourth  va  va,  the  fifth  va  gha  ghata,  the  sixth 
va  gha,  the  seventh  va  va  gha  ghata  {ghata  signifies 
addition).  The  irrational  square  root  is  called  karana 
or  ka.  In  the  £ulvasutras,  which  are  classed  among 
the  religious  books  of  the  Hindus,  but  which  in  addi- 
tion contain  certain  arithmetic  and  geometric  deduc- 
tions, the  word  karana  appears  in  conjunction  with 
numerals;  dvikarani=-V^,  trikarani=V&,  da$akarani 
=  1/10.  If  several  unknown  quantities  are  to  be  dis- 
tinguished, the  first  is  called  ya  ;  the  others  are  named 
after  the  colors:  kalaka  or  ka  (black),  nilaka  or  ni 
(blue),  pitaka  or//  (yellow);  for  example,  by  ya  kabha 
is  meant  the  quantity  x-y,  since  bhavita  or  bha  indi- 
cates multiplication.  There  is  also  a  word  for  ' '  equal " ; 
but  as  a  rule  it  is  not  used,  since  the  mere  placing  of 
a  number  under  another  denotes  their  equality. 

In  the  extension  of  the  domain  of  numbers  to  in- 
clude negative  quantities  the  Hindus  were  certainly 


73 


successful.  They  used  them  in  their  calculations, 
and  obtained  them  as  roots  of  equations,  but  never 
regarded  them  as  proper  solutions.  Bhaskara  was 
even  aware  that  a  square  root  can  be  both  positive 
and  negative,  and  also  that  V — a  does  not  exist  for 
the  ordinary  number-system.  He  says  :  "The  square 
of  a  positive  as  well  as  of  a  negative  number  is  posi- 
tive, and  the  square  root  of  a  positive  number  is 
double,  positive,  and  negative.  There  can  be  no 
square  root  of  a  negative  number,  for  this  is  no 
square."* 

The  fundamental  operations  of  the  Hindus,  of 
which  there  were  six,  included  raising  to  powers  and 
extracting  roots.  In  the  extraction  of  square  and  cube 
roots  Aryabhatta  used  the  formulas  for  (a-{-£)2  and 
(a  -\-  £)3,  and  he  was  aware  of  the  advantage  of  sepa- 
rating the  number  into  periods  of  two  and  three  fig- 
ures each,  respectively.  Aryabhatta  called  the  square 
root  varga  mula,  and  the  cube  root  ghana  mula  (mula, 
root,  used  also  of  plants).  Transformations  of  ex- 
pressions involving  square  roots  were  also  known. 
Bhaskara  applied  the  formulaf 


=  V\  (a  +  VcP  —  b)  -f  I/I  (a  — i/a2  —  b  )  , 

and  was  also  able  to  reduce  fractions  with  square  roots 
in  the  denominator  to  forms  having  a  rational  denomi- 
nator. In  some  cases  the  approximation  methods  for 
square  root  closely  resemble  those  of  the  Greeks. 

*  Cantor,  I.,  p.  585.  t  Cantor,  I.,  p.  586. 


74  HISTORY  OF  MATHEMATICS. 

Problems  in  transpositions,  of  which  only  a  few 
traces  are  found  among  the  Greeks,  occupy  consider- 
able attention  among  the  Indians.  Bhaskara  made 
use  of  formulae  for  permutations  and  combinations* 
with  and  without  repetitions,  and  he  was  acquainted 
with  quite  a  number  of  propositions  involving  the 
theory  of  numbers,  which  have  reference  to  quadratic 
and  cubic  remainders  as  well  as  to  rational  right- 
angled  triangles.  But  it  is  noticeable  that  we  discover 
among  the  Indians  nothing  concerning  perfect,  ami- 
cable, defective,  or  redundant  numbers.  The  knowl- 
edge of  figurate  numbers,  which  certain  of  the  Greek 
schools  cultivated  with  especial  zeal,  is  likewise  want- 
ing. On  the  contrary,  we  find  in  Aryabhatta,  Brah- 
magupta  and  Bhaskara  summations  of  arithmetic 
series,  as  well  as  of  the  series  I2  +  22  +  32+  .  .  .,  1s 
-J-  28  -f-  33  -{-  .  .  .  The  geometric  series  also  appears  in 
the  works  of  Bhaskara.  As  regards  calculation  with 
zero,  Bhaskara  was  aware  that  -^-  =  00. 

The  Chinese  also  show  in  their  literature  some 
traces  of  arithmetic  investigations ;  for  example,  the 
binomial  coefficients  for  the  first  eight  powers  are 
given  by  Chu  shi  kih  in  the  year  1303  as  an  "old 
method."  There  is  more  to  be  found  among  the 
Arabs.  Here  we  come  at  the  outset  upon  the  name  of 
Al  Khowarazmi,  whose  Algebra,  which  was  probably 
translated  into  Latin  by  ^Ethelhard  of  Bath,  opens 

'Cantor,  I.,  p.  579. 


ALGEBRA.  75 

with  the  words*  "Al  Khowarazmi  has  spoken."  In 
the  Latin  translation  this  name  appears  as  Algoritmi, 
and  to-day  appears  as  algorism  or  algorithm,  a  word 
completely  separated  from  all  remembrance  of  Al  Kho- 
warazmi, and  much  used  for  any  method  of  computa- 
tion commonly  employed  and  proceeding  according 
to  definite  rules.  In  the  beginning  of  the  sixteenth 
century  there  appears  in  a  published  mathematical 
work  a  tl philo sophus  nomine  Algorithmic ', "  a  sufficient 
proof  that  the  author  knew  the  real  meaning  of  the 
word  algorism.  But  after  this,  all  knowledge  of  the 
fact  seems  to  disappear,  and  it  was  not  until  our  own 
century  that  it  was  rediscovered  by  Reinaud  and  Bon- 
compagni.f 

Al  Khowarazmi  increased  his  knowledge  by  study- 
ing the  Greek  and  Indian  models.  A  known  quantity 
he  calls  a  number,  the  unknown  quantity  jidr  (root) 
and  its  square  mal  (power).  In  Al  Karkhi  we  find  the 
expression  kab  (cube)  for  the  third  power,  and  there 
are  formed  from  these  expressions  mal  mal=x*,  mal 
kab '  =  x5,  kab  kab  =  x*,  mal  mal  kab  =  x"1 ,  etc.  He  also 
treats  simple  expressions  with  square  roots,  but  with- 
out arriving  at  the  results  of  the  Hindus.  There  is  a 
passage  in  Omar  Khayyam  from  which  it  is  to  be  in- 
ferred that  the  extraction  of  roots  was  always  per- 
formed by  the  help  of  the  formula  for  (a-}-b}n.  Al 
Kalsadi  J  contributed  something  new  by  the  introduc- 

*  Cantor,  I.,  p.  671. 

t  Jahrbuch  fiber  die  Fortschrttte  der  Mathematik,  1887,  p.  23. 

t Cantor,  I.,  p.  765. 


76  HISTORY  OF  MATHEMATICS. 

tion  of  a  radical  sign.  Instead  of  placing  the  word 
jidr  before  the  number  of  which  the  square  root  was 
to  be  extracted,  as  was  the  custom,  Al  Kalsadi  makes 
use  only  of  the  initial  letter  ^  of  this  word  and  places 
it  over  the  number,  as, 


2  =  1/2,  i2  =  i/2£,  5  = 
Among  the  Eastern  Arabs  the  mathematicians 
who  investigated  the  theory  of  numbers  occupied 
themselves  particularly  with  the  attempt  to  discover 
rational  right-angled  triangles  and  with  the  problem 
of  finding  a  square  which,  if  increased  or  diminished 
by  a  given  number,  still  gives  a  square.  An  anony- 
mous writer,  for  example,  gave  a  portion  of  the  the- 
ory of  quadratic  remainders,  and  Al  Khojandi  also 
demonstrated  the  proposition  that  upon  the  hypoth- 
esis of  rational  numbers  the  sum  of  two  cubes  cannot 
be  another  third  power.  There  was  also  some  knowl- 
edge of  cubic  remainders,  as  is  seen  in  the  applica- 
tion by  Avicenna  of  the  proof  by  excess  of  nines  in 
the  formation  of  powers.  This  mathematician  gives 
propositions  which  can  be  briefly  represented  in  the 
form* 


etc. 

Ibn  al  Banna  has  deductions  of  a  similar  kind  which 
form  the  basis  of  a  proof  by  eights  and  sevens,  "f 

In  the  domain  of  series  the  Arabs  were  acquainted 

*  Cantor,  I.,  p.  712.  t  Cantor,  I.,  p.  759. 


ALGEBRA.  77 

at  least  with  arithmetic  and  geometric  progressions 
and  also  with  the  series  of  squares  and  cubes.  In 
this  field  Greek  influence  is  unmistakable. 

2.    Algebra. 

The  work  of  Ahmes  shows  that  the  Egyptians 
were  possessed  of  equations  of  the  first  degree,  and 
used  in  their  solution  methods  systematically  chosen. 
The  unknown  x  is  called  hau  (heap);  an  equation* 
appears  in  the  following  form :  heap,  its  f ,  its  \,  its 
i,  its  whole,  gives  37,  that  is  \x  + \x-\-  ^x-\-x  =  37. 

The  ancient  Greeks  were  acquainted  with  the  so- 
lution of  equations  only  in  geometrical  form.  No- 
where, save  in  proportions,  do  we  find  developed  ex- 
amples of  equations  of  the  first  degree  which  would 
show  unmistakably  that  the  root  of  a  linear  equation 
with  one  unknown  was  ever  determined  by  the  inter- 
section of  two  straight  lines ;  but  in  the  cases  of  equa- 
tions of  the  second  and  third  degrees  there  is  an 
abundance  of  material.  In  the  matter  of  notation 
Diophantus  makes  the  greatest  advance.  He  calls 
the  coefficients  of  the  unknown  quantity  TrA^os.  If 
there  are  several  unknowns  to  be  distinguished,  he 
makes  use  of  the  ordinal  numbers  :  6  Trpwros  d/3i0/*os,  6 
Sevrepos,  6  rpiros.  An  equation  f  appears  in  his  works 
in  the  abbreviated  form  : 

*  Matthiessen,  Grundzuge  der  antiken  iind  modernen  Algebra  der  littera- 
len  Gleichungen,  1878,  p.  269.     Hereafter  referred  to  as  Matthiessen. 
t  Matthiessen,  p.  269. 


78  HISTORY  OF  MATHEMATICS. 


Diophantus  classifies  equations  not  according  to  the 
degree,  but  according  to  the  number  of  essentially 
distinct  terms.  For  this  purpose  he  gives  definite 
rules  as  to  how  equations  can  be  brought  to  their  sim- 
plest form,  that  is,  the  form  in  which  both  members 
of  the  equation  have  only  positive  terms.  Practical 
problems  which  lead  to  equations  of  the  first  degree 
can  be  found  in  the  works  of  Archimedes  and  Heron  ; 
the  latter  gives  some  of  the  so-called  "fountain  prob- 
lems," which  remind  one  of  certain  passages  in  the 
work  of  Ahmes.  Equations  of  the  se,cond  degree 
were  mostly  in  the  form  of  proportions,  and  this 
method  of  operation  in  the  domain  of  a  geometric 
algebra  was  well  known  to  the  Greeks.  They  un- 
doubtedly understood  how  to  represent  by  geometric 
figures  equations  of  the  form 

v.,.'        y       v_ 

a?'*''  '  ~^x^"V'y^  ~m' 
where  all  quantities  are  linear.  Every  calculation  of 
means  in  two  equal  ratios,  i.  e.,  in  a  proportion,  was 
really  nothing  more  than  the  solution  of  an  equation. 
The  Pythagorean  school  was  acquainted  with  the 
arithmetic,  the  geometric  and  the  harmonic  means  of 
two  quantities ;  that  is,  they  were  able  to  solve  geo- 
metrically the  equations 

a  +  b  2at 

-*  =  ab,    x= — — . 


2 

According  to  Nicomachus,  Philolaus  called  the  cube 


79 


with  its  six  surfaces,  its  eight  corners,  and  its  twelve 
edges,  the  geometric  harmony,  because  it  presented 
equal  measurements  in  all  directions  ;  from  this  fact, 
it  is  said,  the  terms  "harmonic  mean"  and  "harmo- 
nic proportion"  were  derived,  the  relationship  being: 


=  -=-,  whence  8  = 


2-6-12    . 


i.  e.,  X-. 


6  +  12' 

The  number  of  distinct  proportions  was  later  in- 
creased to  ten,  although  nothing  essentially  new  was 
gained  thereby.  Euclid  gives  thorough  analyses  of 
proportions,  that  is,  of  the  geometric  solution  of  equa- 
tions of  the  first  degree  and  of  incomplete  quadratics ; 
these,  however,  are  not  given  as  his  own  work,  but  as 
the  result  of  the  labors  of  Eudoxus. 

The  solution  of  the  equation  of  the  second  degree 
by  the  geometric  method  of  applying  areas,  largely 
employed  by  the  ancients,  especially  by  Euclid,  de- 
serves particular  attention. 

In  order  to  solve  the  equation 

*2  +  0*  =  £« 

by  Euclid's  method,  the  problem  must  first  be  put  in 
the  following  form  : 

A       E  B      ff 


D 


C 


K      G 


80  HISTORY  OF  MATHEMATICS. 

"To  the  segment  AB  —  a  apply  the  rectangle  DH 
of  known  area  =  £2,  in  such  a  way  that  CH  shall  be  a 
square."  The  figure  shows  that  for  CK=^,  FH= 
xi  _|_  2x  •  f.  -f  (f  ) 2  =  P  +  (f-) 2 ;  but  by  the  Pythagorean 
proposition,  P  -f-  (^-) 2  =  c*,  whence  EH=.c  =  ^-\- x, 
from  which  we  have  x  =  c — y.  The  solution  obtained 
by  applying  areas,  in  which  case  the  square  root  is 
always  regarded  as  positive,  is  accordingly  nothing 
more  than  a  constructive  representation  of  the  value 


In  the  same  manner  Euclid  solves  all  equations  of 
the  form 


and  he  remarks  in  passing  that  where  V  ft*  —  (y)2» 
according  to  our  notation,  appears,  the  condition  for 
a  possible  solution  is  £>^-.  Negative  quantities  are 
nowhere  considered  ;  but  there  is  ground  for  inferring 
that  in  the  case  of  two  positive  solutions  the  Greeks 
regarded  both  and  that  they  also  applied  their  method 
of  solution  to  quadratic  equations  with  numerical  co- 
efficients.* By  applying  their  knowledge  of  propor- 
tion, they  were  able  to  solve  not  only  equations  of  the 
form  x1  ±  ax  ±  b  =  0,  but  also  of  the  more  general 
form 


for  a  as  the  ratio  of  two  line-segments.     Apollonius 

*Zenthen,  Die  Lehre  von  den  Kegelschnitten  im  Altertum.    Dentsch  von 
v.  Fischer-Benzon.     1886. 


accomplished  this  with  the  aid  of  a  conic,  having  the 
equation 


The  Greeks  were  accordingly  able  to  solve  every  gen- 
eral equation  of  the  second  degree  having  two  essen- 
tially different  coefficients,  which  might  also  contain 
numerical  quantities,  and  to  represent  their  positive 
roots  geometrically. 

The  three  principal  forms  of  equations  of  the  sec- 
ond degree  first  to  be  freed  from  geometric  statement 
and  completely  solved,  are 


The  solution  consisted  in  applying  an  area,  the  prob- 
lem being  to  apply  to  a  given  line  a  rectangle  in  such 
a  manner  that  it  would  either  contain  a  given  area  or 
be  greater  or  less  than  this  given  area  by  a  constant. 
For  these  three  conditions  there  arose  the  technical 
expressions  Trapa^oAiy,  vTrep/JoAiy,  e\Aeu/ns,  which  after 
Archimedes  came  to  refer  to  conies.* 

In  later  times,  with  Heron  and  Diophantus,  the 
solution  of  equations  of  the  second  degree  was  partly 
freed  from  the  geometric  representation,  and  passed 
into  the  form  of  an  arithmetic  computation  proper 
(while  disregarding  the  second  sign  in  the  square 
root). 

The  equation  of  the  third  degree,  owing  to  its 
dependence  on  geometric  problems,  played  an  im- 

*  Tannery  in  Bordeaux  Mem.,  IV. 


82  HISTORY  OF  MATHEMATICS. 

portant  part  among  the  Greeks.  The  problem  of  the 
duplication  (and  also  the  multiplication)  of  the  cube 
attained  especial  celebrity.  This  problem  demands 
nothing  more  than  the  solution  of  the  continued  pro- 
portion a:x  =  x:y=y:2a,  that  is,  of  the  equation 
x*  =  2a?  (in  general  xz  =  ^a?).  This  problem  is  very 
old  and  was  considered  an  especially  important  one 
by  the  leading  Greek  mathematicians.  Of  this  we 
have  evidence  in  a  passage  of  Euripides  in  which  he 
makes  King  Minos  say  concerning  the  tomb  of  Glau- 
cus  which  is  to  be  rebuilt*:  "  The  enclosure  is  too 
small  for  a  royal  tomb  :  double  it,  but  fail  not  in  the 
cubical  form."  The  numerous  solutions  of  the  equa- 
tion x3  =  2as  obtained  by  Hippocrates,  Plato,  Me- 
naechmus,  Archytas  and  others,  followed  the  geomet- 
ric form,  and  in  time  the  horizon  was  so  considerably 
extended  in  this  direction  that  Archimedes  in  the 
study  of  sections  of  a  sphere  solved  equations  of  the 
form 


by  the  intersection  of  two  lines  of  the  second  degree, 
and  in  doing  so  also  investigated  the  conditions  to  be 
fulfilled  in  order  that  there  should  be  no  root  or  two 
or  three  roots  between  0  and  a.  Since  the  method 
of  reduction  by  means  of  which  Archimedes  obtains 
the  equation  x8  —  axi-±-flc  =  Q  can  be  applied  with 
considerable  ease  to  all  forms  of  equations  of  the  third 
degree,  the  merit  of  having  set  forth  these  equations 

*  Cantor,  I.,  p.  199. 


ALGEBRA.  83 

in  a  comprehensive  manner  and  of  having  solved  one 
of  their  principal  groups  by  geometric  methods  be- 
longs without  question  to  the  Greeks.* 

We  find  the  first  trace  of  indeterminate  equations 
in  the  cattle  problem  {Problema  bovinutn)  of  Archi- 
medes. 

This  problem,  which  was  published  in  the  year  1773  by  Les- 
sing,  from  a  codex  in  the  library  at  Wolfenbiittel,  as  the  first  of 
four  unprinted  fragments  of  Greek  anthology,  is  given  in  twenty- 
two  distichs.  In  all  probability  it  originated  directly  with  Archi- 
medes who  desired  to  show  by  means  of  this  example  how,  pro- 
ceeding from  simple  numerical  quantities,  one  could  easily  arrive 
at  very  large  numbers  by  the  interweaving  of  conditions.  The 
problem  runs  something  as  follows :  f 

The  sun  had  a  herd  of  bulls  and  cows  of  different  colors,  (i) 
Of  Bulls  the  white  ( IV)  were  in  number  (\  -f  £)  of  the  black  (X) 
and  the  yellow  (F);  the  black  (X)  were  (£  +  £)  of  the  dappled  (Z) 
and  the  yellow  '(F);  the  dappled  (Z)  were  (J+f)  of  the  white 
( IV}  and  the  yellow  ( Y).  (2)  Of  Cows  which  had  the  same  colors 
(vt.x.y.z),  «/  =  (4 +  J)  (*  +  *),  *  =  (i  +  J)(-Z  +  *),  *  =  (*  +  i) 
(Y+y),  *  =  (i  +  *)(»r+«').  W+^is  to  be  a  square;  F+Z 
a  triangular  number. 

The  problem  presents  nine  equations  with  ten  unknowns  : 


*Zeuthen,  Die  Lehre  von  den  Kegelschnitten  im  Alterturn.     Deutsch  von 
.  Fischer-Benzon      1886. 

t  Krumbiegel  und  Amthor,  Das  Problema  bovinum  des  Archimedes.    Schlo- 
'Mi's  Zeitschrift,  Bd.  25,  HI.  A.;   Gow,  p.  99. 


84  HISTORY  OF  MATHEMATICS. 

According  to  Amthor  the  solution  is  obtained  by  Pell's  equation 
P — 2 '3 '7 '11 '29 '353  «2  =  1,  assuming  the  condition  w  =  0  (mod. 
2 '4657),  in  which  process  there  arises  a  continued  fraction  with  a 
period  of  ninety-one  convergents.  If  we  omit  the  last  two  condi- 
tions, we  get  as  the  total  number  of  cattle  5916837175686,  a 
number  which  is  nevertheless  much  smaller  than  that  involved  in 
the  sand-reckoning  of  Archimedes. 

But  the  name  of  Diophantus  is  most  closely  con- 
nected with  systems  of  equations  of  this  kind.  He 
endeavors  to  satisfy  his  indeterminate  equations  not 
by  means  of  whole  numbers,  but  merely  by  means  of 
rational  numbers  (always  excluding  negative  quanti- 
ties) of  the  form  —  where  p  and  a  must  be  positive  in- 

q 

tegers.  It  appears  that  Diophantus  did  not  proceed 
in  this  field  according  to  general  methods,  but  rather 
by  ingeniously  following  out  special  cases.  At  least 
those  of  his  solutions  of  indeterminate  equations  of 
the  first  and  second  degrees  with  which  we  are  ac- 
quainted permit  of  no  other  inference.  Diophantus 
seems  to  have  been  not  a  little  influenced  by  earlier 
works,  such  as  those  of  Heron  and  Hypsicles.  It  may 
therefore  be  assumed  that  even  before  the  Christian 
era  there  existed  an  indeterminate  analysis  upon 
which  Diophantus  could  build.* 

The  Hindu  algebra  reminds  us  in  many  respects 
of  Diophantus  and  Heron.  As  in  the  case  of  Dio- 
phantus, the  negative  roots  of  an  equation  are  not 
admitted  as  solutions,  but  they  are  consciously  set 

*P.  Tannery,  in  Mtmoires  de  Bordeaux,  1880.  This  view  of  Tannery's  is 
controverted  by  Heath,  T.  L.,  Diofhantos  of  Alexandria^  1885,  p.  135. 


ALGEBRA.  85 

aside,  which  marks  an  advance  upon  Diophantus. 
The  transformation  of  equations,  the  combination  of 
terms  containing  the  same  powers  of  the  unknown,  is 
also  performed  as  in  the  works  of  Diophantus.  The 
following  is  the  representation  of  an  equation  accord- 
ing to  Bhaskara  :* 

va  va  2  I  va  1  I  ru  30 

— ,  i.  e., 
va  va  0  |  va  0  |  ru    8 

Equations  of  the  first  degree  appear  not  only  with 
one,  but  also  with  several  unknowns.  The  Hindu 
method  of  treating  equations  of  the  second  degree 
shows  material  advance.  In  the  first  place,  ax^  -f  bx 
=  c  is  considered  the  only  typef  instead  of  the  three 
Greek  forms  ax*-{-bx  =  c)  bx-\-c=ax^,  ax"2  -f  c  =  bx. 
From  this  is  easily  derived  4a2^2  -{-  4al>x  =  4ac,  and 
then  (2ax  -\-  3)2  =  kac  -f  ^2,  whence  it  follows  that 


Bhaskara  goes  still  further.  He  considers  both  signs 
of  the  square  root  and  also  knows  when  it  cannot  be 
extracted.  The  two  values  of  the  root  are,  however, 
admitted  by  him  as  solutions  only  when  both  are  posi- 
tive,— evidently  because  his  quadratic  equations  ap- 
pear exclusively  in  connection  with  practical  problems 
of  geometric  form.  Bhaskara  also  solves  equations 
of  the  third  and  fourth  degrees  in  cases  where  these 

*Matthiessen,  p.  269.  t  Cantor,  I.,  p.  585. 


86  HISTORY  OF  MATHEMATICS. 

equations  can  be  reduced  to  equations  of  the  second 
degree  by  means  of  advantageous  transformations  and 
the  introduction  of  auxiliary  quantities. 

The  indeterminate  analysis  of  the  Hindus  is  espe- 
cially prominent.  Here  in  contrast  to  Diophantus 
only  solutions  in  positive  integers  are  admitted.  In- 
determinate equations  of  the  first  degree  with  two  or 
more  unknowns  had  already  been  solved  by  Arya- 
bhatta,  and  after  him  by  Bhaskara,  by  a  method  in 
which  the  Euclidean  algorism  for  finding  the  greatest 
common  divisor  is  used ;  so  that  the  method  of  solu- 
tion corresponds  at  least  in  its  fundamentals  with  the 
method  of  continued  fractions.  Indeterminate  equa- 
tions of  the  second  degree,  for  example  those  of  the 
form  xy  =  ax  -f-  by  -f-  c,  are  solved  by  arbitrarily  as- 
signing a  value  to  y  and  then  obtaining  x,  or  geo- 
metrically by  the  application  of  areas,  or  by  a  cyclic 
method.*  This  cyclic  method  does  not  necessarily 
lead  to  the  desired  end,  but  may  nevertheless,  by  a 
skilful  selection  of  auxiliary  quantities,  give  integ- 
ral values.  It  consists  in  solving  in  the  first  place, 
instead  of  the  equation  ax*  -\-  b  =  cy*,  the  equation 
ax2  -j- 1  =y*.  This  is  done  by  the  aid  of  the  empiri- 
cally assumed  equation  a^3  +  ^=(72,  from  which 
other  equations  of  the  same  form,  aA^n-\- £„  =  C%,  can 
be  deduced  by  the  solution  of  indeterminate  equations 
of  the  first  degree.  By  means  of  skilful  combinations 

•Cantor,  I.,  p.  591. 


the   equations   aA2n-{- £„=(%   furnish   a   solution   of 
ax*-\-l=y'** 

The  algebra  of  the  Chinese,  at  least  in  the  earliest 
period,  has  this  in  common  with  the  Greek,  that  equa- 
tions of  the  second  degree  are  solved  geometrically. 
In  later  times  there  appears  to  have  been  developed 
a  method  of  approximation  for  determining  the  roots 
of  higher  algebraic  equations.  For  the  solution  of  in- 
determinate equations  of  the  first  degree  the  Chinese 
developed  an  independent  method.  It  bears  the  name 
of  the  "great  expansion"  and  its  discovery  is  ascribed 
to  Sun  tse,  who  lived  in  the  third  century  A.  D.  This 
method  can  best  be  briefly  characterised  by  the  fol- 
lowing example :  Required  a  number  x  which  when 
divided  by  7, 11,  15  gives  respectively  the  remainders 
2,  5,  7.  Let  k\,  ki,  k%.  be  found  so  that 

11-15-*!  15-7-,*2 

— ir-    =  ?i  +  i-      — -  =  V*  +  TT> 


15 

we  have,  for  example,  k\  =  Z,  &2  =  2,  *8  =  8,  and  ob- 
tain the  further  results 

11-15-2=330,         330-2=   660, 

15-   7-2  =  210,         210-5  =  1050, 

7-11-8  =  616,         616-7  =  4312, 

660  +  1050  +  4312  =  6022;       ~~^  =5+  -^^  > 
^  =  247  is  then  a  solution  of  the  given  equation,  f 

•Cantor,  I.,  p.  593.       tL.  MatlhiesseninSMomzIcA'sZeztscAnyt,  XXVI. 


88  HISTORY  OF  MATHEMATICS. 

In  the  writing  of  their  equations  the  Chinese  make 
as  little  use  as  the  Hindus  of  a  sign  of  equality.  The 
positive  coefficients  were  written  in  red,  the  negative 
in  black.  As  a  rule  tde  is  placed  beside  the  absolute 
term  of  the  equation  and  yuen  beside  the  coefficient  of 
the  first  power  ;  the  rest  can  be  inferred  from  the  ex- 
ample 14.x3  —  27#  =  17,*  where  r  and  b  indicate  the 
color  of  the  coefficient : 

r!4         or         r!4         or         r!4 

roo  roo  roo 

,27  621yuen 

r\1tae  r!7. 

The  Arabs  were  pupils  of  both  the  Hindus  and 
the  Greeks.  They  made  use  of  the  methods  of  their 
Greek  and  Hindu  predecessors  and  developed  them, 
especially  in  the  direction  of  methods  of  calculation. 
Here  we  find  the  origin  of  the  word  algebra  in  the 
writings  of  Al  Khowarazmi  who,  in  the  title  of  his 
work,  speaks  of  "al-jabr  wo1 1  muqabalah"  i.  e.,  the 
science  of  redintegration  and  equation.  This  expres- 
sion denotes  two  of  the  principal  operations  used  by 
the  Arabs  in  the  arrangement  of  equations.  When 
from  the  equation  Xs -\-  r  =  x* -{- px -{- r  the  new  equa- 
tion x3  =  x2  -\-px  is  formed,  this  is  called  al  nmqabalah; 
the  transformation  which  gives  from  the  equation 
px  —  g=x*  the  equation  px=x"*-\-g,  a  transforma- 
tion which  was  considered  of  great  importance  by  the 

*  Cantor,  I.,  p.  643. 


ALGEBRA.  89 

ancients,  was  called  al-jabr,  and  this  name  was  ex- 
tended to  the  science  which  deals  in  general  with 
equations. 

The  earlier  Arabs  wrote  out  their  equations  in 
words,  as  for  example,  Al  Khowarazmi*  (in  the  Latin 
translation)  : 

Census  et  quinque  radices  equantur  viginti  quatuor 
x>      +  5*  =  24; 

and  Omar  Khayyam, 

Cubus,  latera  et  numerus  aequales  sunt  quadratis 
xs    +  bx    -f         c  ace*. 

In  later  times  there  arose  among  the  Arabs  quite  an 
extended  symbolism.  This  notation  made  the  most 
marked  progress  among  the  Western  Arabs.  The 
unknown  x  was  called  jidr,  its  square  mal;  from  the 
initials  of  these  words  they  obtained  the  abbrevia- 
tions x=:(J&,  x2  =  -o.  Quantities  which  follow  di- 
rectly one  after  another  are  added,  but  a  special  sign 
is  used  to  denote  subtraction.  "  Equals  "  is  denoted 
by  the  final  letter  of  adala  (equality),  namely,  by  means 
of  a  final  lam.  In  Al  Kalsadif  3#2=12#  +  63  and 
^x^-\-x  =  1^  are  represented  by 


and  the  proportion  7  :  12  =  84  :  x  is  given  the  form 
^.-.84.  -.12.  -.7. 

*Matthiessen,  p.  269.  t  Cantor,  I.,  p.  767. 


90  HISTORY  OF  MATHEMATICS. 

Diophantus  had  already  classified  equations,  not 
according  to  their  degree,  but  according  to  the  num- 
ber of  their  terms.  This  principle  of  classification  we 
find  completely  developed  among  the  Arabs.  Ac- 
cording to  this  principle  Al  Khowarazmi*  forms  the 
following  six  groups  for  equations  of  the  first  and 
second  degrees  : 

x2  =  ax  ("a  square  is  equal  to  roots"), 
x2  =  a  ("a  square  is  equal  to  a  constant"), 
ax  =  b,     x*  -\-  ax  =  b,     x^  +  a  =  bx,     ax  -f  b  =  x2, 
("roots  and  a  constant  are  equal  to  a  square"). 

The  Arabs  knew  how  to  solve  equations  of  the  first 
degree  by  four  different  methods,  only  one  of  which 
has  particular  interest,  and  that  because  in  modern 
algebra  it  has  been  developed  as  a  method  of  approx- 
imation for  equations  of  higher  degree.  This  method 
of  solution,  Hindu  in  its  origin,  is  found  in  particular 
in  Ibn  al  Banna  and  Al  Kalsadi  and  is  there  called 
the  method  of  the  scales.  It  went  over  into  the  Latin 
translations  as  the  regula  falsorum  and  regula  falsi.  To 
illustrate,  let  the.  equation  ax-}-  6  =  0  be  givenf  and 
let  z\  and  z%  be  any  numerical  quantities  ;  then  if  we 
place  az\  -\-  b  =y\,  az^ 


y\  —yi 

as  can  readily  be  seen.  Ibn  al  Banna  makes  use  of 
the  following  graphic  plan  for  the  calculation  of  the 
value  of  x  : 

*  Matthiessen,  p.  270.  t  Matthiessen,  p.  277. 


The  geometric  representation,  which  with  y  as  a  neg- 
ative quantity  somewhat  resembles  a  pair  of  scales, 
would  be  as  follows,  letting  O#1=z1,  0^  =  2^,  B\C\ 
y9,  OA=x: 

Gz 


0      B, 


From  this  there  results  directly 


x~ 
x— 


that  is,  that  the  errors  in  the  substitutions  bear  to 
each  other  the  same  ratio  as  the  errors  in  the  results, 
the  method  apparently  being  discovered  through  geo- 
metric considerations. 

In  the  case  of  equations  of  the  second  degree  Al 
Khowarazmi  gives  in  the  first  place  a  purely  mechan- 
ical solution  (negative  roots  being  recognised  but  not 
admitted),  and  then  a  proof  by  means  of  a  geometric 
figure.  He  also  undertakes  an  investigation  of  the 
number  of  solutions.  In  the  case  of 

x*  +  c  =  bxy  from  which  x  =  £  =fc  1/(|)2  —  *» 
Al  Khowarazmi  obtains  two  solutions,  one  or  none 
according  as 


92 


HISTORY  OF  MATHEMATICS. 


(!)2>'>    (*)'=<.    &<<• 

He  gives  the  geometric  proof  for  the  correctness  of 
the  solution  of  an  equation  like  x*  -\-2x  =  ~L5,  where 
he  takes  x  =  3,  in  two  forms,  either  by  means  of  a 
perfectly  symmetric  figure,  or  by  the  gnomon.  In 
the  first  case,  for  A£  =  x,  BC=%,  BD  =  \,  we  have 
A 


G 

*2  +  4-i-*  +  4-(£)2  =  15  +  l,  O-fl)2  =  16;  in  the 
second  we  have  #2-f  2-1- x  +  I2  =  15  -f  1.  In  the 
treatment  of  equations  of  the  form  ax*"  ±  bxn  d=  c  =  0, 
the  theory  of  quadratic  equations  receives  still  further 
development  at  the  hands  of  Al  Kalsadi. 

Equations  of  higher  degree  than  the  second,  in 
the  form  in  which  they  presented  themselves  to  the 
Arabs  in  the  geometric  or  stereometric  problems  of 
the  Greek  type,  were  not  solved  by  them  arithmeti- 
cally, but  only  by  geometric  methods  with  the  aid 
of  conies.  Here  Omar  Khayyam*  proceeded  most 
systematically.  He  solved  the  following  equations 
of  the  third  degree  geometrically : 


'Matthiessen,  pp  367,  894,  945. 


ALGEBRA.  Q3 


r=qx, 


The  following  is  the  method  of  expression  which  he 
employs  in  these  cases  : 

"A  cube  and  square  are  equal  to  roots;" 
"a  cube  is  equal  to  roots,  squares  and  one  number," 
when  the  equations 

x3  +PX*  =  qx,     xs  =px*  -\-qx-\-r 

are  to  be  expressed.  Omar  calls  all  binomial  forms 
simple  equations  ;  trinomial  and  quadrinomial  forms 
he  calls  composite  equations.  He  was  unable  to  solve 
the  latter,  even  by  geometric  methods,  in  case  they 
reached  the  fourth  degree. 

The  indeterminate  analysis  of  the  Arabs  must  be 
traced  back  to  Diophantus.  In  the  solution  of  inde- 
terminate equations  of  the  first  and  second  degree 
Al  Karkhi  gives  integral  and  fractional  numbers,  like 
Diophantus,  and  excludes  only  irrational  quantities. 
The  Arabs  were  familiar  with  a  number  of  proposi- 
tions in  regard  to  Pythagorean  triangles  without  hav- 
ing investigated  this  field  in  a  thoroughly  systematic 
manner. 

C.   THE  SECOND  PERIOD. 

TO  THE  MIDDLE  OF  THE  SEVENTEENTH  CENTURY. 

As  long  as  the  cultivation  of  the  sciences  among 
the  Western  peoples  was  almost  entirely  confined  to 


94  HISTORY  OF  MATHEMATICS. 

the  monasteries,  during  a  period  lasting  from  the 
eighth  century  to  the  twelfth,  no  evidence  appeared 
of  any  progress  in  the  general  theory  of  numbers. 
As  in  the  learned  Roman  world  after  the  end  of  the 
fifth  century,  so  now  men  recognised  seven  liberal 
arts, — the  trivium,  embracing  grammar,  rhetoric  and 
dialectics,  and  the  quadriviwn,  embracing  arithmetic, 
geometry,  music  and  astronomy.*  But  through  Arab 
influence,  operating  in  part  directly  and  in  part 
through  writings,  there  followed  in  Italy  and  later  also 
in  France  and  Germany  a  golden  age  of  mathemat- 
ical activity  whose  influence  is  prominent  in  all  the 
literature  of  that  time.  Thus  Dante,  in  the  fourth 
canto  of  the  Divina  Commedia  mentions  among  the 
personages 

"...  who  slow  their  eyes  around 

Majestically  moved,  and  in  their  port 

Bore  eminent  authority," 

a  Euclid,  a  Ptolemy,  a  Hippocrates  and  an  Avicenna. 
There  also  arose,  as  a  further  development  of  cer- 
tain famous  cloister,  cathedral  and  chapter  schools, 
and  in  rare  instances,  independent  of  them,  the  first 
universities,  at  Paris,  Oxford,  Bologna,  and  Cam- 
bridge, which  in  the  course  of  the  twelfth  century 
associated  the  separate  faculties,  and  from  the  begin- 
ning of  the  thirteenth  century  became  famous  as  Stu- 
dia  generalia."\  Before  long  universities  were  also  es- 

*Muller,  Historisch-ttymologische  Studien  Vber  mathtntatische  Tertnino- 
logie,  1887. 

t  Suter,  Die  Mathematik  an/den  Universitiiten  des  Mittelalters,  1887. 


95 


tablished  in  Germany  (Prague,  1348;  Vienna,  1365; 
Heidelberg,  1386  ;  Cologne,  1388  ;  Erfurt,  1392  ;  Leip- 
zig, 1409;  Rostock,  1419;  Greifswalde,  1456;  Basel, 
1459;  Ingolstadt,  1472;  Tubingen  and  Mainz,  1477), 
in  which  for  a  long  while  mathematical  instruction 
constituted  merely  an  appendage  to  philosophical  re- 
search. We  must  look  upon  Johann  von  Gmunden  as 
the  first  professor  in  a  German  university  to  devote 
himself  exclusively  to  the  department  of  mathematics. 
From  the  year  1420  he  lectured  in  Vienna  upon  mathe- 
matical branches  only,  and  no  longer  upon  all  depart- 
ments of  philosophy,  a  practice  which  was  then  uni- 
versal. 

I.    General  Arithmetic. 

Even  Fibonacci  made  use  of  words  to  express 
mathematical  rules,  or  represented  them  by  means  of 
line-segments.  On  the  other  hand,  we  find  that  Luca 
Pacioli,  who  was  far  inferior  to  his  predecessor  in 
arithmetic  inventiveness,  used  the  abbreviations  ./., 
.m.,  J?.  for  plus,  minus,  and  radix  (root).  As  early  as 
1484,  ten  years  before  Pacioli,  Nicolas  Chuquet  had 
written  a  work,  in  all  probability  based  upon  the  re- 
searches of  Oresme,  in  which  there  appear  not  only 
thfc  signs  p  and  m  (for  plus  and  minus),  but  also  ex- 
pressions like 

I£4.10,  $2.17  for  V\§,  1/17. 
He  also   used  the  Cartesian  exponent-notation,  and 


g6  HISTORY  OF  MATHEMATICS. 

the  expressions  equipolence,  equipolent,  for  equivalence 
and  equivalent.* 

Distinctively  symbolic  arithmetic  was  developed 
upon  German  soil.  In  German  general  arithmetic 
and  algebra,  in  the  Deutsche  Coss,  the  symbols  -f-  and 
—  for  plus  and  minus  are  characteristic,  f  They  were 
in  common  use  while  the  Italian  school  was  still  writ- 
ing /  and  m.  The  earliest  known  appearance  of  these 
signs  is  in  a  manuscript  {Regula  Cose  vel  Algebre}  of 
the  Vienna  library,  dating  from  the  middle  of  the  fif- 
teenth century.  In  the  beginning  of  the  seventeenth 
century  Reymers  and  Faulhaber  used  the  sign  -j-,| 
and  Peter  Roth  the  sign  -H-  as  minus  signs. 

Among  the  Italians  of  the  thirteenth  and  four- 
teenth centuries,  in  imitation  of  the  Arabs,  the  course 
of  an  arithmetic  operation  was  expressed  entirely  in 
words.  Nevertheless,  abbreviations  were  gradually  in- 
troduced and  Luca  Pacioli  was  acquainted  with  such 
abbreviations  to  express  the  first  twenty-nine  powers 
of  the  unknown  quantity.  In  his  treatise  the  absolute 
term  and  x,  x2,  x3,  x*,  x5,  x6,  .  .  .  are  always  respec- 
tively represented  by  numero  or  n",  cosa  or  co,  censo  or 
ce,  cube  or  cu,  censo  de  censo  or  ce.ce,  primo  relate  or 
p.'r",  censo  de  cuba  or  ce.cu  .  .  . 

The  Germans  made  use  of  symbols  of  their  o.wn 

*A.  Marre  in  Boncompagni ' s  Bulletino,  XIII.  Jahrbuch  uber  die  Fort- 
tchritte  der  Math.,  1881,  p.  8. 

tTreutlein,  "Die  deutsche  Coss,"  Schliimilch's  Zeitschrift,  Bd.  24,  HI.  A. 
Hereafter  referred  to  as  Treutlein. 

tThe  sign  -s-  is  first  used  as  a  sign  of  division  in  Rahn's  Teutsche  Algebra. 
Zurich,  1659. 


ALGEBRA.  97 

invention.  Rudolff  and  Riese  represented  the  abso- 
lute term  and  the  powers  of  the  unknown  quantity  in 
the  following  manner :  Dragma,  abbreviated  in  writ- 
ing, <f>;  radix  (or  coss,  i.  e.,  root  of  the  equation)  is 
expressed  by  a  sign  resembling  an  r  with  a  little  flour- 
ish ;  zensus  by  5 ;  cubus  by  c  with  a  long  flourish  on 
top  in  the  shape  of  an  /  (in  the  following  pages  this 
will  be  represented  merely  by  c);  zensus  de  zensu  (zens- 
dezens)  by  33,  sursolidum  by  (3  or  j| ;  zensikubus  by  $c ; 
bissursolidum  by  bif  or  3f ;  zensus  zensui  de  zensu  (zens- 
zensdezens)  by  553 ;  cubus  de  cubo  by  cc. 

There  are  two  opinions  concerning  the  origin  of 
the  x  of  mathematicians.  According  to  the  one,  it  was 
originally  an  r  (radix)  written  with  a  flourish  which 
gradually  came  to  resemble  an  x,  while  the  original 
meaning  was  forgotten,  so  that  half  a  century  after 
Stifel  it  was  read  by  all  mathematician^  as  x.*  The 
other  explanation  depends  upon  the  fact  that  it  is  cus- 
tomary in  Spain  to  represent  an  Arabic  s  by  a  Latin 
x  where  whole  words  and  sentences  are  in  question  ; 
for  instance  the  quantity  I2x,  in  Arabic  G"  is  repre- 
sented by  12  xai,  more  correctly  by  12  sat.  Accord- 
ing to  this  view,  the  x  of  the  mathematicians  would 
be  an  abbreviation  of  the  Arabic  sai=xat,  an  expres- 
sion for  the  unknown  quantity. 

By  the  older  cossistsf  these  abbreviations  are  in- 
troduced without  any  explanation ;  Stifel,  however, 

*Treutlein.     G.  Wertheim  in  Schlomilch's  Zeitschri/t,  Bd.  44,  HI.  A. 
tTreutlein. 


98  HISTORY  OF  MATHEMATICS. 

considers  it  necessary  to  give  his  readers  suitable  ex- 
planations. The  word  "root,"  used  for  the  first  power 
of  an  unknown  quantity  he  explains  by  means  of 
the  geometric  progression,  "because  all  successive 
members  of  the  series  develop  from  the  first  as  from 
a  root"  ;  he  puts  for  #°,  xl,  x1,  Xs,  x*,  .  .  .  the  signs  1, 
Ix,  lj,  lc,  155,  .  .  .  and  calls  these  "  cossic  numbers," 
which  can  be  continued  to  infinity,  while  to  each  is 
assigned  a  definite  order-number,  that  is,  an  expo- 
nent. In  the  German  edition  of  Rudolff' s  Coss,  Stifel 
at  first  writes  the  "cossic  series"  to  the  seventeenth 
power  in  the  manner  already  indicated,  but  also  later 
as  follows: 

1  .   ik      12UI.      laaut      UOnUt    etc. 

He  also  makes  use  of  the  letters  3  and  (£  in  writing 
this  expression.  The  nearest  approach  to  our  present 
notation  is  to  be  found  in  Burgi  and  Reymers,  where 
with  the  aid  of  "exponents"  or  "characteristics"  the 
polynomial  8*«  +  I2x5  —  9**  +  10.*3  +  3*2  +  1x  —  4  is 
represented  in  the  following  manner  : 

VI  V  IV  III  II  I  O 

8  +  12  —  9  +  10  +  3  +  7  —  4 

In  Scheubel  we  find  for  x,  x*,  xs,  x*,  x6 .  .  . ,  pri, , 
sec.,  ter.t  quar.,  quin.,  and  in  Ramus  /,  q,  c,  bq,  s  as 
abbreviations  for  latus,  quadratus,  cubus,  biquadratits, 
solidus. 

The  product  <7*«  —  3*  +  2)  (5*  —  3)  =  85*8  —  36*2 
+  19#  —  6  is  represented  in  its  development  by  Gram- 
mateus,  Stifel,  and  Ramus  in  the  following  manner  : 


9rJ 


GKAMMATEUS  :  STIFEL  : 

lx.  __  %pri.  +  2N  7  5  —  3*  -f  2 

by      $pri.  —  ZN  5*  —  3 


35^—153  +  10* 
—  21*.+    9pri.  —  QN  —  215+    9*—  6 

35  /^.—  36*.  -j-  19/r/.  —  6^V  35<r  —  363  +  19#—  6 

RAMUS  : 

2 
—  3 


9/—  6 


—  6. 

x\s  early  as  the  fifteenth  century  the  German  Coss 
made  use  of  a  special  symbol  to  indicate  the  extrac- 
tion of  the  root.  At  first  .4  was  used  for  1/4  ;  this 
period  placed  before  the  number  was  soon  extended 
by  means  of  a  stroke  appended  to  it.  Riese  and 
Rudolff  write  merely  j/4  for  T/4.  Stifel  takes  the 
first  step  towards  a  more  general  comprehension  of 
radical  quantities  in  his  Arithmetica  Integra,  where  the 
second,  third,  fourth,  fifth,  roots  of  six  are  represented 
,  yV6,  1/336,  |/f  6,  while  elsewhere  the  symbols 


V. 


are  used  as  radical  signs.  These  symbols,  of  which 
the  first  two  occur  in  Rudolff  and  the  other  three  in  a 
work  of  Stifel,  indicate  respectively  the  third,  fourth, 
second,  third,  and  fourth  roots  of  the  numbers  which 
they  precede. 


100  HISTORY  OF  MATHEMATICS. 

Rudolff  gives  a  few  rules  for  operations  with  rad- 
ical quantities,  but  without  demonstrations.  Like 
Fibonacci  he  calls  an  irrational  number  a  numerus 
surdus.  Such  expressions  as  the  following  are  intro- 
duced : 

I/a  ±  Vb  = 


—  I/O  +  £)2 


_          _ 
Va±.Vb  a  —  b 

Stifel  enters  upon  the  subject  of  irrational  numbers 
with  especial  zeal  and  even  refers  to  the  speculations 
of  Euclid,  but  preserves  in  all  his  developments  a 
well-grounded  independence.  Stifel  distinguishes  two 
classes  of  irrational  numbers  :  principal  and  subordi- 
nate (Haupt-  und  Nebenarteri).  In  the  first  class  are 
included  (1)  simple  irrational  numbers  of  the  form 
v/iz,  (2)  binomial  irrationals  with  the  positive  sign,  as 

l/SS^  +  l/SSfi,  4  +  ^6,  J/J12  +  JA12, 
(3)  square  roots  of  such  binomial  irrationals  as 

V*  •  1/36  +  1/38  = 


(4)  binomial  irrationals  with  the  negative  sign,  as 
1/5510  —  1/536»  and  (5)  square  roots  of  such  binomial 
irrationals,  as 


^3  •  1/36  —  1/38  =     1/6  —  1/8. 

The  subordinate  class  of  irrational  quantities,  accord- 
ing to  Stifel,  includes  expressions  like 


ALGEBRA. 


Fibonacci  evidently  obtained  his  knowledge  of 
negative  quantities  from  the  Arabs,  and  like  them  he 
does  not  admit  negative  quantities  as  the  roots  of  an 
equation.  Pacioli  enunciates  the  rule,  minus  times 
minus  gives  always  plus,  but  he  makes  use  of  it  only 
for  the  expansion  of  expressions  of  the  form  (/  —  ^) 
(r  —  j).  Cardan  proceeds  in  the  same  way;  he  recog- 
nises negative  roots  of  an  equation,  but  he  calls  them 
aestimationes  falsae  or  fictae,*  and  attaches  to  them  no 
independent  significance.  Stifel  calls  negative  quan- 
tities numeri  absurdi.  Harriot  is  the  first  to  consider 
negative  quantities  in  themselves,  allowing  them  to 
form  one  side  of  an  equation.  Calculations  involving 
negative  quantities  consequently  do  not  begin  until 
the  seventeenth  century.  It  is  the  same  with  irratio- 
nal numbers  ;  Stifel  is  the  first  to  include  them  among 
numbers  proper. 

Imaginary  quantities  are  scarcely  mentioned.  Car- 
dan incidentally  proves  that 

(5  4.  i/^T5)  -  (5  —  1/^15)  =40. 
Bombelli    goes   considerably  farther.     Although   not 
entering  into  the  nature  of  imaginary  quantities,  of 
which   he   calls   -f-  V  —  1  piu  di  meno,   and  —  l/  —  1 
meno  di  meno,  he  gives  rules  for  the  treatment  of  ex- 

*  Ars  mag-tia,  1545.    Cap.  I.,  6. 


IO2  HISTORY  OF  MATHEMATICS. 

pressions  of  the  form  a-\-bV  —  1,  as  they  occur  in 
the  solution  of  the  cubic  equation. 

The  Italian  school  early  made  considerable  ad- 
vancement in  calculations  involving  powers.  Nicole 
Oresme*  had  long  since  instituted  calculations  with 
fractional  exponents.  In  his  notation 


it  appears  that  he  was  familiar  with  the  formulae 


i     i 


A-      JL 


In  the  transformation  of  roots  Cardan  made  the  first 
important  advance  by  writing 


and  therefore  ta2  —  b=p*  —  g  =  c,  a*  —  b  =  cz.    Bom- 
bellif  enlarged  upon  this  observation  and  wrote 

V  a  +  V  —  b—p-\-V  —  g,      V  a  —  V  —  b=p  —  I/  —  q, 
from  which  follows  ^cP  +  b  =p'i  -\-  q.    With  reference 


to  the  equation  x3  =  15.*  +  4  he  discovered  that 


For  in  this  case 


become  through  addition  /*  —  3/^  =  2,  and  with  q  = 
5  —  /*,  4/3  —  15/  =  2,  and  consequently  (by  trial)/  =  2 
and  =  1. 


Hankel,  p.  350. 


t  Cantor,  II.,  p.  572. 


ALGEBRA.  103 

The  extraction  of  square  and  cube  roots  accord- 
ing to  the  Arab,  or  rather  the  Indian,  method,  was  set 
forth  by  Grammateus.  In  the  process  of  extracting 
the  square  root,  for  the  purpose  of  dividing  the  num- 
ber into  periods,  points  are  placed  over  the  first,  third, 
fifth,  etc.,  figures,  counting  from  right  to  left.  Stifel* 
developed  the  extracting  of  roots  to  a  considerable 
extent;  it  is  undoubtedly  for  this  purpose  that  he 
worked  out  a  table  of  binomial  coefficients  as  far  as 
O  +  £)17,  in  which,  for  example,  the  line  for  (a+£)4 
reads : 

15S     .     4         6         4         1     . 

The  theory  of  series  in  this  period  made  no  ad- 
vance upon  the  knowledge  of  the  Arabs.  Peurbach 
found  the  sum  of  the  arithmetic  and  the  geometric 
progressions.  Stifel  examined  the  series  of  natural 
numbers,  of  even  and  of  odd  numbers  and  deduced 
from  them  certain  power  series.  In  regard  to  these 
series  he  was  familiar,  through  Cardan,  with  the  the- 
orem that  l  +  2  +  22  +  2»  +  . .  +  2'-1  =  2«— 1.  With 
Stifel  geometric  progressions  appear  in  an  application 
which  is  not  found  in  Euclid's  treatment  of  means,  f 
As  is  well  known,  n  geometric  means  are  inserted  be- 
tween the  two  quantities  a  and  b  by  means  of  the 
equations 

—  —  —  —  —  ==        =  XH~*  =  -f  =  a 
xi  ~~  *2  ~  *s  ~~  xn    ~~  b  ~ 

•Cantor,  II.,  pp.  397,  409.  tTrentlein. 


i  o4 


HISTORY  OF  MATHEMATICS. 


where  ^  =  "+t/Jl.  Stifel  inserts  five  geometric  means 
between  the  numbers  6  and  18  in  the  following  man- 
ner : 

1  3  9  27  81  243  729 


6 


18 


in  which  the  last  line  is  obtained  from  the  preceding 
by  multiplying  by  6.  Stifel  makes  use  of  this  solution 
for  the  purpose  of  duplicating  the  cube.  He  selects  6 
for  the  edge  of  the  given  cube  ;  three  geometric  means 
are  to  be  inserted  between  6  and  12,  and  as  q  =  i//%, 
the  edge  of  the  required  cube  will  be  x  =  6f/2  = 
l/<:432.  This  length  is  constructed  geometrically  by 
Stifel  in  the  following  manner  : 


In  the  right  angled  triangle  ACJ3,  with  the  hypotenuse 
BC,  let  A£  =  Q,  AC=12;  make  AD  =  DC,  AE=- 
ED,  AF=FE,  FJ=JE,  JK=JC=JL.  Then  AK 
is  the  first,  AL  the  second  geometric  mean  between 
6  and  12.  This  construction,  which  Stifel  regards  as 
entirely  correct,  is  only  an  approximation,  since 
AK=1.§  instead  of  6  ^2  =  7.56,  AL  =  3l/TO  =  9 .487 
instead  of  6^4  =  9. 524. 


ALGEBRA. 


105 


Simple  facts  involving  the  theory  of  numbers  were 
also  known  to  Stifel,  such  as  theorems  relating  to 
perfect  and  diametral  numbers  and  to  magic  squares. 

A  diametral  number  is  the  product  of  two  numbers 
the  sum  of  whose  squares  is  a  rational  square,  the 
square  of  the  diameter,  e.  g.,  652  =  252  +  602  =  392-f 
522,  and  hence  25.60  =  1500  and  39.52  =  2028  are 
diametral  numbers  of  equal  diameter. 


11 

24 

7 

20 

3 

4 

12 

25 

8 

16 

17 

5 

13 

21 

9 

10 

18 

1 

14 

22 

23 

6 

19 

2 

15 

Magic  squares  are  figures  resembling  a  chess 
board,  in  which  the  terms  of  an  arithmetic  progres- 
sion are  so  arranged  that  their  sum,  whether  taken 
diagonally  or  by  rows  or  columns,  is  always  the  same. 
A  magic  square  containing  an  odd  number  of  cells, 
which  is  easier  to  construct  than  one  containing  an 
even  number,  can  be  obtained  in  the  following  man- 
ner :  Place  1  in  the  cell  beneath  the  central  one,  and 
the  other  numbers,  in  their  natural  order,  in  the  empty 
cells  in  a  diagonal  direction.  Upon  coming  to  a  cell 


IO6  HISTORY  OF  MATHEMATICS. 

already  occupied,  pass  vertically  downwards  over  two 
cells.*  Possibly  magic  squares  were  known  to  the  Hin- 
dus ;  but  of  this  there  is  no  certain  evidence,  f  Manuel 
MoschopulusJ  (probably  in  the  fourteenth  century) 
touched  upon  the  subject  of  magic  squares.  He  gave 
definite  rules  for  the  construction  of  these  figures, 
which  long  after  found  a  wider  diffusion  through  La- 
hire  and  Mollweide.  During  the  Middle  Ages  magic 
squares  formed  a  part  of  the  wide-spread  number- 
mysticism.  Stifel  was  the  first  to  investigate  them  in 
a  scientific  way,  although  Adam  Riese  had  already  in- 
troduced the  subject  into  Germany,  but  neither  he  nor 
Riese  was  able  to  give  a  simple  rule  for  their  con- 
struction. We  may  nevertheless  assume  that  towards 
the  end  of  the  sixteenth  century  such  rules  were  known 
to  a  few  German  mathematicians,  §  as  for  instance,  to 
the  Rechenmeister  of  Nuremberg,  Peter  Roth.  In  the 
year  1612  Bachet  published  in  his  Problemes  plaisants  || 
a  general  rule  for  squares  containing  an  odd  number 
of  cells,  but  acknowledged  that  he  had  not  succeeded 
in  finding  a  solution  for  squares  containing  an  even 
number.  Fre"nicle  was  the  first  to  make  a  real  ad- 
vance beyond  Bachet.  He  gave  rules  (1693)  for  both 
classes  of  squares,  and  even  discovered  squares  that 
maintain  their  characteristics  after  striking  off  th 

*  Unger,  p.  109. 

t  Montucla,  Histoire  des  MatMmatiques,  1799-1802. 
J  Cantor,  I.,  p.  480. 

§Giesing,  Leben  und  Schriftcn  Leonardo's  da  Pisa,  1886. 
|  This  work  is  now  accessible  in  a  new  edition  published  in  1884,  Paris. 
Gauthier-Villars. 


ALGEBRA.  IOJ 

outer  rows  and  columns.  In  1816  Mollweide  collected 
the  scattered  rules  into  a  book,  De  quadratis  magicis, 
which  is  distinguished  by  its  simplicity  and  scientific 
form.  More  modern  works  are  due  to  Hugel  (Ans- 
bach,  1859),  to  Pessl  (Amberg,  1872),  who  also  con- 
siders a  magic  cylinder,  and  to  Thompson  {Quarterly 
Journal  of  Mathematics,  Vol.  X.),  by  whose  rules  the 
magic  square  with  the  side  pn  is  deduced  from  the 
square  with  the  side  #.* 

2.  Algebra. 

Towards  the  end  of  the  Middle  Ages  the  Ars  major, 
Arte  maggiore,  Algebra  or  the  Coss  is  opposed  to  the 
ordinary  arithmetic  (Ars  minor).  The  Italians  called 
the  theory  of  equations  either  simply  Algebra,  like  the 
Arabs,  or  Ars  magna,  Ars  rei  et  census  (very  common 
after  the  time  of  Leonardo  and  fully  settled  in  Regio- 
montanus),  La  regola  della  cosa  (cosa-=res,  thing), 
Ars  cossica  or  Regula  cosae.  The  German  algebraists 
of  the  fifteenth  and  sixteenth  centuries  called  it  Coss, 
Regula  Coss,  Algebra,  or,  like  the  Greeks,  Logistic. 
Vieta  used  the  term  Arithmetica  speciosa,  and  Reymers 
Arithmetica  analytica,  giving  the  section  treating  of 
equations  the  special  title  von  der  Aequation.  The 
method  of  representing  equations  gradually  took  on 
the  modern  form.  Equality  was  generally,  even  by 
the  cossists,  expressed  by  words  ;  it  was  not  until  the 
middle  of  the  seventeenth  century  that  a  special  sym- 

*Gnnther,  "Ueber  magische  Quadrate,"  Grunert's  Arch.,  Bd.  57. 


I08  HISTORY  OF  MATHEMATICS. 

bol  came  into  common  use.     The  following  are  exam- 
ples of  the  different  methods  of  representing  equa- 
tions :* 
Cardan  : 

Cubus  /  6  rebus  aequalis  20,  x*  +    6#  =  20  ; 

Vieta  : 

1C—8Q  +  1QN  aequ.40,  *»  —  8.x2  +  16*  =  40  ; 
Regiomontanus  : 

16  census  et  2000  aequ.  680  rebus,  16#2  +  2000  = 

680*; 
Reymers  : 

XXVIII    XII  X  VI  III  I  O 

1^65532       +18      -=-30     -5-  18     +12    -4-8; 
*28:=  65532^2  +  18*10  —  30*6  — 
Descartes  : 


0,     /  —  8/— 
^.6  *    *    *    *  _bx  x0,     x*  —  bx 

X5    *     *     *      *    _b       x0>        ^5_^ 

Hudde  : 

xsx>  qx  .r,  xz  =  qx  +  r. 

In  Euler's  time  the  last  transformation  in  the  develop- 
ment of  the  modern  form  had  already  been  accom- 
plished. 

Equations  of  the  first  degree  offer  no  occasion  for 
remark.  We  may  nevertheless  call  attention  to  the 
peculiar  form  of  the  proportion  which  is  found  in 
Grammateus  and  Apian,  f  The  former  writes  :  "Wie 

*  Matthiessen,  GrttndzKge  der  antiken  und  modernen  Algebra,  2  ed.,  1896, 
p.  270,  etc. 

t  Gerhardt,  Geschichte  der  Mathematik  in  Dewtschland,  1877. 


ALGEBRA.  IOQ 

sich  hadt  a  zum  b,  also  hat  sich  c  zum  d,"  and  the 
latter  places 

4-12-9-0  for  A  =  A. 

Leonardo  of  Pisa  solved  equations  of  the  second  de- 
gree in  identically  the  same  way  as  the  Arabs.*  Car- 
dan recognized  two  roots  of  a  quadratic  equation,  even 
when  one  of  them  was  negative;  but  he  did  not  regard 
such  a  root  as  forming  an  actual  solution.  Rudolff 
recognized  only  positive  roots,  and  Stifel  stated  ex- 
plicitly that,  with  the  exception  of  the  case  of  quad- 
ratic equations  with  two  positive  roots,  no  equation 
can  have  more  than  one  root.  In  general,  the  solu- 
tion was  affected  in  the  manner  laid  down  by  Gram- 
mateusf  in  the  example  12.x -f  24  =  2i§*2:  "Proceed 
thus:  divide  2>±N  by  21°.  sec.,  which  gives  10f<z 
(10f  =  a).  Also  divide  12  pri.  by  2io  sec.,  which 
gives  the  result  5|£(5|  =  £).  Square  the  half  of  b, 
which  gives  &£££,  to  which  add  a  =  10f,  giving  -659^, 
of  which  the  square  root  is  -J^.  Add  this  to  £  of  b,  or 
if ,  and  7  is  the  number  represented  by  1  pfi.  Proof : 
12X1N=%±N;  add  24^,  =  108^.  2J£  sec.  multi- 
plied by  49  must  also  give  108 A"." 

This  "German  Coss "  was  certainly  cultivated  by 
Hans  Bernecker  in  Leipzig  and  by  Hans  Conrad  in 
EislebenJ  (about  1525),  yet  no  memoranda  by  either 
of  these  mathematicians  have  been  found.  The  Uni- 
versity of  Vienna  encouraged  Gramrnateus  to  publish, 

*  Cantor,  II.,  p.  31.  t  Gerhardt.  J  Cantor,  II.,  p.  387. 


HO  HISTORY  OF  MATHEMATICS. 

in  the  year  1523,  the  first  German  treatise  on  Algebra 
under  the  title,  "  Eyn  new  kunstlich  behend  vnd  geiviss 
Rechenbilchlin  \  vff  alle  Kauffmannschafft.  Nach  Ge- 
meynen  Regeln  de  tre.  Welschen  practic.  Regeln 
falsi.  Etlichen  Regeln  Cosse  .  .  Buchhalten  .  .  Visier 
Ruthen  zu  machen."  Adam  Riese,  who  had  pub- 
lished his  Arithmetic  in  1518,  completed  in  1524  the 
manuscript  of  the  Coss ;  but  it  remained  in  manu- 
script and  was  not  found  until  1855  in  Marienberg. 
The  Coss  published  by  Christoff  Rudolff  in  1525  in 
Strassburg  met  with  universal  favor.  This  work, 
which  is  provided  with  many  examples,  all  completely 
solved,  is  described  in  the  following  words : 

1 '  Behend  vnd  Hiibsch  Rechnung  durch  die  kunstreichen  re- 
geln  Algebre  |  so  gemeinicklich  die  Coss  genennt  werden.  Darinnen 
alles  so  treulich  an  Tag  geben  |  das  auch  allein  ausz  vleissigem 
Lesen  on  alien  miindtliche  vnterricht  mag  begriffen  werden.  Hind- 
angesetzt  die  meinung  aller  dere  |  so  bisher  vil  vngegriindten  regeln 
angehangen.  Einem  jeden  liebhaber  diser  kunst  lustig  vnd  ergetz- 
lich  Zusamen  bracht  durch  Christoffen  Rudolff  von  Jawer."* 

The  principal  work  of  the  German  Coss  is  Michael 
Stifel's  Arithmetica  Integra,  published  in  Nuremberg  in 
1544.  In  this  book,  besides  the  more  common  opera- 
tions of  arithmetic,  not  only  are  irrational  quantities 
treated  at  length,  but  there  are  also  to  be  found  appli- 

*A  translation  would  read  somewhat  as  follows:  "Rapid  and  neat  com- 
putation by  means  of  the  ingenious  rules  of  algebra,  commonly  designated 
the  Coss.  Wherein  are  faithfully  elucidated  all  things  in  such  wise  that  they 
may  be  comprehended  from  diligent  reading  alone,  without  any  oral  instruc- 
tion whatsoever.  In  disregard  of  the  opinions  of  all  those  who  hitherto  have 
adhered  to  numerous  unfounded  rules.  Happily  and  divertingly  collected 
'or  lovers  of  this  art,  by  Christoff  Rudolff,  of  Jauer." 


cations  of  algebra  to  geometry.  Stifel  also  published 
in  1553  Die  Cosz  Christoffs  Rudolfs  mil  schonen  Ex- 
empeln  der  Cosz  Gebessert  vnd  sehr  gemehrt,  with  copi- 
ous appendices  of  his  own,  giving  compendia  of  the 
Coss.  With  pardonable  self-appreciation  Stifel  as- 
serts, "It  is  my  purpose  in  such  matters  (as  far  as  I 
am  able)  from  complexity  to  produce  simplicity. 
Therefore  from  many  rules  of  the  Coss  I  have  formed 
a  single  rule  and  from  the  many  methods  for  roots 
have  also  established  one  uniform  method  for  the  in- 
numerable cases." 

Stifel's  writings  were  laid  under  great  contribu- 
tion by  later  writers  on  mathematics  in  widely  distant 
lands,  usually  with  no  mention  of  his  name.  This 
was  done  in  the  second  half  of  the  sixteenth  century 
by  the  Germans  Christoph  Clavius  and  Scheubel,  by 
the  Frenchmen  Ramus,  Peletier,  and  Salignac,  by 
the  Dutchman  Menher,  and  by  the  Spaniard  Nunez. 
It  can,  therefore,  be  said  that  by  the  end  of  the  six- 
teenth century  or  the  beginning  of  the  seventeenth 
the  spirit  of  the  German  Coss  dominated  the  Algebra 
of  all  the  European  lands,  with  the  single  exception 
of  Italy. 

The  history  of  the  purely  arithmetical  solution  of 
equations  of  the  third  and  fourth  degrees  which  was 
successfully  worked  out  upon  Italian  soil  demands 
marked  attention.  Fibonacci  (Leonardo  of  Pisa)* 
made  the  first  advance  in  this  direction  in  connection 

*  Cantor,  II.,  p.  43. 


112  HISTORY  OF  MATHEMATICS. 

with  the  equation  x*  +  2*2  +  10*  =  20.  Although  he 
succeeded  in  solving  this  only  approximately,  it  fur- 
nished him  with  the  opportunity  of  proving  that  the 
value  of  x  cannot  be  represented  by  square  roots 
alone,  even  when  the  latter  are  chosen  in  compound 
form,  like 


The  first  complete  solution  of  the  equation  x*-{-mx=n 
is  due  to  Scipione  del  Ferro,  but  it  is  lost.*  The 
second  discoverer  is  not  Cardan,  but  Tartaglia.  On 
the  twelfth  of  February,  1535,  he  gave  the  formula 
for  the  solution  of  the  equation  x3-\-mx  =  n,  which 
has  since  become  so  famous  under  the  name  of  his 
rival.  By  1541  Tartaglia  was  able  to  solve  any  equa- 
tion whatsoever  of  the  third  degree.  In  1539  Cardan 
enticed  his  opponent  Tartaglia  to  his  house  in  Milan 
and  importuned  him  until  the  latter  finally  confided 
his  method  under  the  pledge  of  secrecy.  Cardan  broke 
his  word,  publishing  Tartaglia's  solution  in  1545  in 
his  Ars  magna,  although  not  without  some  mention  of 
the  name  of  the  discoverer.  Cardan  also  had  the  satis- 
faction of  giving  to  his  contemporaries,  in  his  Ars 
magna,  the  solution  of  the  biquadratic  equation  which 
his  pupil  Ferrari  had  succeeded  in  obtaining.  Bom- 
belli  is  to  be  credited  with  representing  the  roots  of 
the  equation  of  the  third  degree  in  the  simplest  form, 
in  the  so-called  irreducible  case,  by  means  of  a  trans- 
formation of  the  irrational  quantities.  Of  the  German 

*Hankel,  p.  360. 


ALGEBRA.  113 

mathematicians,  Rudolff  also  solved  a  few  equations 
of  the  third  degree,  but  without  explaining  the  method 
which  he  followed.  Stifel  by  this  time  was  able  to 
give  a  brief  account  of  the  "cubicoss,"  that  is,  the 
theory  of  equations  of  the  third  degree  as  given  in 
Cardan's  work.  The  first  complete  exposition  of  the 
Tartaglian  solution  of  equations  of  the  third  degree 
comes  from  the  pen  of  Faulhaber  (1604). 

The  older  cossists*  had  arranged  equations  of  the 
first,  second,  third,  and  fourth  degrees  (in  so  far  as 
they  allow  of  a  solution  by  means  of  square  roots 
alone)  in  a  table  containing  twenty-four  different 
forms.  The  peculiar  form  of  these  rules,  that  is,  of 
the  equations  with  their  solutions,  can  be  seen  in  the 
following  examples  taken  from  Riese  : 

"The  first  rule  is  when  the  root  [of  the  equation] 
is  equal  to  a  number,  or  dragma  so  called.  Divide 
by  the  number  of  roots  ;  the  result  of  this  division 
must  answer  the  question."  (I.  e.,  if  ax  =  b,  then 

b 

x  =  — , 
a 

"The  sixteenth  rule  is  when  squares  equal  cubes 
and  fourth  powers.  Divide  through  by  the  number 
of  fourth  powers  [the  coefficient  of  #4],  then  take  half 
the  number  of  cubes  and  multiply  this  by  itself,  add 
this  product  to  the  number  of  squares,  extract  the 
square  root,  and  from  the  result  take  half  the  number 
of  cubes.  Then  you  have  the  answer." 

*Treutlein. 


114  HISTORY  OF  MATHEMATICS. 

Taking  this  step  by  step  we  have, 

ax*  +  by?  =  ex*,    x*  -I  --  x*  =  —  x2,  or 
a  a 


The  twenty-four  forms  of  the  older  cossists  are  re- 
duced by  Riese  to  "acht  equationes"  (eight  equa- 
tions, as  his  combination  of  German  and  Latin  means), 
but  as  to  the  fact  that  the  square  root  is  two-valued 
he  is  not  at  all  clear.  Stifel  was  the  first  to  let  a  single 
equation  stand  for  these  eight,  and  he  expressly  as 
serts  that  a  quadratic  can  have  only  two  roots  ;  this 
he  asserts,  however,  only  for  the  equation  x2  =ax  —  b. 
In  order  to  reduce  the  equations  above  mentioned  to 
one  of  Riese's  eight  forms,  Rudolff  availed  himself  of 
"  four  precautions  (Cautelen),"  from  which  it  is  clearly 
seen  what  labor  it  cost  to  develop  the  coss  step  by 
step.  For  example,  here  is  his 

'*  First  precaution.  When  in  equating  two  num- 
bers, in  the  one  is  found  a  quantity,  and  in  the  other 
is  found  one  of  the  same  name,  then  (considering  the 
signs  -f-  and  —  )  must  one  of  these  quantities  be  added 
to  or  subtracted  from  the  other,  one  at  a  time,  care 
being  had  to  make  up  for  the  defect  in  the  equated 
numbers  by  subtracting  the  +  and  adding  the  —  ." 
(I.  e.,  from  5*8  —  3*  +  4  =  2#8  +  5#,  we  derive  3#2-f 
4  =  8*.) 

The  first  examples  of  this  period,  of  equations 
with  more  than  one  unknown  quantity,  are  met  with 


ALGEBRA.  I  15 

in  Rudolff,*  who  treats  them  only  incidentally.  Here 
also  Stifel  went  decidedly  beyond  his  predecessors. 
Besides  the  first  unknown,  Ix,  he  introduced  1A,  l£, 
1C,  ...  as  secundae  radices  or  additional  unknowns 
and  indicated  the  new  notation  made  necessary  in  the 
performance  of  the  fundamental  operations,  as  8xA 
(=8xy},  lA$(=y*),  and  several  others. 

Cardan,  over  whose  name  a  shadow  has  been  cast 
by  his  selfishness  in  his  intercourse  with  Tartaglia,  is 
still  deserving  of  credit,  particularly  for  his  approxi- 
mate solution  of  equations  of  higher  degrees  by  means 
of  the  regula  falsi  which  he  calls  regula  aurea.  Vieta 
went  farther  in  this  direction  and  evolved  a  method 
of  approximating  the  solution  of  algebraic  equations 
of  any  degree  whatsoever,  the  method  improved  by 
Newton  and  commonly  ascribed  to  him.  Reymers  and 
Biirgi  also  contributed  to  these  methods  of  approxi- 
mation, using  the  regula  falsi.  We  can  therefore  say 
that  by  the  beginning  of  the  seventeenth  century  there 
were  practical  methods  at  hand  for  calculating  the 
positive  real  roots  of  algebraic  equations  to  any  de- 
sired degree  of  exactness. 

The  real  theory  of  algebraic  equations  is  especially 
due  to  Vieta.  He  understood  (admitting  only  posi- 
tive roots)  the  relation  of  the  coefficients  of  equations 
of  the  second  and  third  degree  to  their  roots,  and  also 
made  the  surprising  discovery  that  a  certain  equation 
of  the  forty-fifth  degree,  which  had  arisen  in  trig- 

*  Cantor,  II.,  p.  392. 


Il6  HISTORY  OF  MATHEMATICS. 

onometric  work,  possessed  twenty-three  roots  (in  this 
enumeration  he  neglected  the  negative  sine).  In  Ger- 
man writings  there  are  also  found  isolated  statements 
concerning  the  analytic  theory  of  equations ;  for  ex- 
ample, Burgi  recognized  the  connection  of  a  change 
of  sign  with  a  root  of  the  equation.  However  unim- 
portant these  first  approaches  to  modern  theories  may 
appear,  they  prepared  the  way  for  ideas  which  be- 
came dominant  in  later  times. 

D.    THIRD  PERIOD. 

FROM  THE  MIDDLE  OF  THE  SEVENTEENTH  CENTURY  TO 
THE  PRESENT  TIME. 

The  founding  of  academies  and  of  royal  societies 
characterizes  the  opening  of  this  period,  and  is  the 
external  sign  of  an  increasing  activity  in  the  field  of 
mathematical  sciences.  The  oldest  learned  society, 
the  Accademia  dei  Lincei,  was  organized  upon  the 
suggestion  of  a  Roman  gentleman,  the  Duke  of  Cesi. 
as  early  as  1603,  and  numbered,  among  other  famous 
members,  Galileo.  The  Royal  Society  of  London  was 
founded  in  1660,  the  Paris  Academy  in  1666,  and  the 
Academy  of  Berlin  in  1700.* 

With  the  progressive  development  of  pure  mathe- 
matics the  contrast  between  arithmetic,  which  has  to 
do  with  discrete  quantities,  and  algebra,  which  relates 
rather  to  continuous  quantities,  grew  more  and  more 

*  Cantor,  III.,  pp.  7,  29. 


ALGEBRA.  II J 

marked.  Investigations  in'  algebra  as  well  as  in  the 
theory  of  numbers  attained  in  the  course  of  time  great 
proportions. 

The  mighty  impulse  which  Vieta's  investigations 
had  given  influenced  particularly  the  works  of  Har- 
riot. Building  upon  Vieta's  discoveries,  he  gave  in 
his  Artis  analyticae  praxis,  published  posthumously  in 
the  year  1631,  a  theory  of  equations,  in  which  the  sys- 
tem of  notation  was  also  materially  improved.  The 
signs  >  and  <  for  "greater  than"  and  "less  than" 
originated  with  Harriot,  and  he  always  wrote  x*  for 
xx  and  xz  for  xxx,  etc.  The  sign  X  for  "times" 
is  found  almost  simultaneously  in  both  Harriot  and 
Oughtred,  though  due  to  the  latter ;  Descartes  used 
a  period  to  indicate  multiplication,  while  Leibnitz  in 
1686  indicated  multiplication  by  ^-N  and  division  by "— ', 
although  already  in  the  writings  of  the  Arabs  the  quo- 
tient of  a  divided  by  b  had  appeared  in  the  forms 
a  —  b,  a/b,  or—.  The  form  a\b  is  used  for  the  first 
time  by  Clairaut  in  a  work  which  was  published  post- 
humously in  the  year  1760.  Wallis  made  use  in  1655 
of  the  sign  oo  to  indicate  infinity.  Descartes  made  ex- 
tensive use  of  the  the  form  a"  (for  positive  integral  ex- 
ponents). Wallis  explained  the  expressions  x~"  and 
x*  as  indicating  the  same  thing  as  l:x*  and  v^x  re- 
spectively j  but  Leibnitz  and  Newton  were  the  first  to 
recognize  the  great  importance  of,  and  to  suggest,  a 
consistent  system  of  notation. 


Il8  HISTORY  OF  MATHEMATICS. 

The  powers  of  a  binomial  engaged  the  attention 
of  Pascal  in  his  correspondence  with  Fermat  in  1654,* 
which  contains  the  "arithmetic  triangle,"  although, 
in  its  essential  nature  at  least,  it  had  been  suggested 
by  Stifel  more  than  a  hundred  years  before.  This 
arithmetic  triangle  is  a  table  of  binomial  coefficients 
arranged  in  the  following  form : 


1 

1 

1 

1 

1 

1 

1 

2 

3 

4 

5 

6 

1 

3 

6 

10 

15 

21 

1 

4 

10 

20 

35 

56 

1 

5 

15 

35 

70 

126 

1 

6 

21 

56 

126 

252 

so  that  the  nth  diagonal,  extending  upwards  from  left 
to  right  contains  the  coefficients  of  the  expansion  of 
(a  _j_  £)«.  Pascal  used  this  table  for  developing  figurate 
numbers  and  the  combinations  of  a  given  number  of 
elements.  Newton  generalized  the  binomial  formula 
in  1669,  Vandermonde  gave  an  elementary  proof  in 
1764,  and  Euler  in  1770  in  his  Anleitung  zur  Algebra 
gave  a  proof  for  any  desired  exponent. 

A  series  of  interesting  investigations,  for  the  most 
part  belonging  to  the  second  half  of  the  nineteenth 
century,  relates  to  the  nature  of  number  and  the  ex- 
tension of  the  number- concept.  While  among  the  an- 
cients a  "number"  meant  one  of  the  series  of  natural 

*  Cantor,  II.,  p.  684. 


ALGEBRA.  IIQ 

numbers  only,  in  the  course  of  time  the  fundamental 
operations  of  arithmetic  have  been  extended  from 
whole  to  fractional,  from  positive  to  negative,  from 
rational  and  real  to  irrational  and  imaginary  numbers. 
For  the  addition  of  natural,  or  integral  absolute, 
numbers,  which  by  Newton  and  Cauchy  are  often 
termed  merely  "numbers,"  the  associative  and  com- 
mutative laws  hold  true,  that  is, 


Their  multiplication  obeys  the  associative,  commuta- 
tive, and  distributive  laws,  so  that 

abc^=(ab)c\  ab  =  ba;  (a  -f-  b)  c  =  ac  +  be. 

To  these  direct  operations  correspond,  as  inverses, 
subtraction  and  division.  The  application  of  these 
operations  to  all  natural  numbers  necessitates  the  in- 
troduction of  the  zero  and  of  negative  and  fractional 
numbers,  thus  forming  the  great  domain  of  rational 
numbers,  within  which  these  operations  are  always 
valid,  if  we  except  the  one  case  of  division  by  zero. 

This  extension  of  the  number-system  showed  itself 
in  the  sixteenth  century  in  the  introduction  of  negative 
quantities.  Vieta  distinguished  affirmative  (positive) 
and  negative  quantities.  But  Descartes  was  the  first 
to  venture,  in  his  geometry,  to  use  the  same  letter  for 
both  positive  and  negative  quantities. 

The  irrational  had  been  incorporated  by  Euclid 
into  the  mathematical  system  upon  a  geometric  basis, 
this  plan  being  followed  for  many  centuries.  Indeed 


120  HISTORY  OF  MATHEMATICS. 

it  was  not  until  the  most  modern  times*  that  a  purely 
arithmetic  theory  of  irrational  numbers  was  produced 
through  the  researches  of  Weierstrass,  Dedekind,  G. 
Cantor,  and  Heine. 

Weierstrass  proceeds  f  from  the  concept  of  the 
whole  number.  A  numerical  quantity  consists  of  a 
series  of  objects  of  the  same  kind;  a  number  is  there- 
fore nothing  more  than  the  "combined  representation 
of  one  and  one  and  one,  etc.  "I  By  means  of  subtrac- 
tion and  division  we  arrive  at  negative  and  fractional 
numbers.  Among  the  latter  there  are  certain  numbers 
which,  if  referred  to  one  particular  system,  for  exam- 
ple to  our  decimal  system,  consist  of  an  infinite  num- 
ber of  elements,  but  by  transformation  can  be  made 
equal  to  others  arising  from  the  combination  of  a  finite 
number  of  elements  (e.  g.,  0.1333...=^).  These 
numbers  are  capable  of  still  another  interpretation. 
But  it  can  be  proved  that  every  number  formed  from 
an  infinite  number  of  elements  of  a  known  species, 
and  which  contains  a  known  finite  number  of  those 
elements,  possesses  a  very  definite  meaning,  whether 
it  is  capable  of  actual  expression  or  not.  When  a 
number  of  this  kind  can  only  be  represented  by  the 
infinite  number  of  its  elements,  and  in  no  other  way, 
it  is  an  irrational  number. 

Dedekind§  arranges  all  positive  and  negative,  in- 

*Stolr,  yorlesungen  iiber  allgctncine  Arithmetik,  1885-1886. 

+  Kossak,  Die  Elemente  der  Aritkmetik,  1872. 

t  Rosier,  Die  neueren  Definitionsformen  der  irrationalen  Zahlen,  1886. 

§  Dedekind.  Stttigkett  und  irrationale  Zaklen,  1872. 


ALGEBRA.  121 

tegral  and  fractional  numbers,  according  to  their  mag- 
nitude, in  a  system  or  in  a  body  of  numbers  (Zahlcn 
korper},  R.  A  given  number,  a,  divides  this  system 
into  the  two  classes,  A\  and  A%,  each  containing  in- 
finitely many  numbers,  so  that  every  number  in  A\  is 
less  than  every  number  in  A%.  Then  a  is  either  the 
greatest  number  in  A\  or  the  least  in  A$.  These  ra- 
tional numbers  can  be  put  into  a  one-to-one  corre- 
spondence with  the  points  of  a  straight  line.  It  is 
then  evident  that  this  straight  line  contains  an  infinite 
number  of  other  points  than  those  which  correspond 
to  rational  numbers,  that  is,  the  system  of  rational 
numbers  does  not  possess  the  same  continuity  as  the 
straight  line,  a  continuity  possible  only  by  the  intro- 
duction of  new  numbers.  According  to  Dedekind  the 
essence  of  continuity  is  contained  in  the  following 
axiom  :  "If  all  the  points  of  a  straight  line  are  divided 
into  two  classes  such  that  every  point  of  the  first  class 
lies  to  the  left  of  every  point  of  the  second,  then  there 
exists  one  point  and  only  one  which  effects  this  divi- 
sion of  all  points  into  two  classes,  this  separation  of 
the  straight  line  into  two  parts."  With  this  assump- 
tion it  becomes  possible  to  create  irrational  numbers. 
A  rational  number,  a,  produces  a  Schnitt  or  section 
(Ai\Aj),  with  respect  to  A\  and  AS,  with  the  charac- 
teristic property  that  there  is  'in  A\  a  greatest,  or  in 
AI  a  least  number,  a.  To  every  one  of  the  infinitely 
many  points  of  the  straight  line  which  are  not  covered 
by  rational  numbers,  or  in  which  the  straight  line  is 


122  HISTORY  OF  MATHEMATICS. 

not  cut  by  a  rational  number,  there  corresponds  one 
and  only  one  section  (Ai\A^),  and  each  one  of  these 
sections  defines  one  and  only  one  irrational  number  a. 

In  consequence  of  these  distinctions  ' '  the  system  R  constitutes 
an  organized  domain  of  all  real  numbers  of  one  dimension;  by  this 
no  more  is  meant  to  be  said  than  that  the  following  laws  govern  :  * 

I.  If  a  >  Si,  and  /3  >  y,  then  a  is  also  >  y ;  that  is,  the  number 
/3  lies  between  the  numbers  a,  y. 

II.  If  a,  y  are  two  distinct  numbers,  then  there  are  infinitely 
many  distinct  numbers  /?  which  lie  between  a  and  y. 

III.  If  a  is  a  definite  number,  then  all  numbers  of  the  system 
R  fall  into  two  classes,  Al  and  A2,  each  of  which  contains  infinitely 
many  distinct  numbers;  the  first  class  At  contains  all  numbers 
al  which  are  <a;  the  second  class  A2  contains  all  numbers  a2 
which  are  >  a  ;  the  number  a  itself  can  be  assigned  indifferently 
to  either  the  first  or  the  second  class  and  it  is  then  respectively 
either  the  greatest  number  of  the  first  class,  or  the  least  of  the  sec- 
ond.    In  every  case,  the  separation  of  the  system  R  into  the  two 
classes  A^  and  A2  is  such  that  every  number  of  the  first  class  A , 
is  less  than  every  number  of  the  second  class  At,  and  we  affirm 
that  this  separation  is  effected  by  the  number  a. 

IV.  If  the  system  R  of  all  real  numbers  is  separated  into  two 
classes,  Alt  A2,  such  that  every  number  alt  of  the  class  A\  is  less 
than  every  number  a2  of  the  class  A2,  then  there  exists  one  and 
only  one  number  a  by  which  this  separation  is  effected  (the  domain 
R  possesses  the  property  of  continuity)." 

According  to  the  assertion  of  J.  Tanneryf  the  fundamental 
ideas  of  Dedekind's  theory  had  already  appeared  in  J.  Bertrand'F 
text-books  of  arithmetic  and  algebra,  a  statement  denied  by  Dede 
kind.J 

*  Dedekind,  Stetigkeit  *nd  irrationale  Zahbn,  1872. 

t  Stolz,  Vorleiungen  uber  allgemeine  Arithmetik,  1885-1886. 

t  Dedekind,  Was  sind  und  was  sallen  die  Zahlenf    1888. 


ALGEBRA.  123 

G.  Cantor  and  Heine*  introduce  irrational  num- 
bers through  the  concept  of  a  fundamental  series. 
Such  a  series  consists  of  infinitely  many  rational  num- 
bers, a\,  a?,  as,  .  .  .  .  an+r,  .  .  .,  and  it  possesses  the 
property  that  for  an  assumed  positive  number  e,  how- 
ever small,  there  is  an  index  n,  so  that  for  «>»i  the 
absolute  value  of  the  difference  between  the  term  an 
and  any  following  term  is  smaller  than  e  (condition  of 
the  convergency  of  the  series  of  the  a's).  Any  two 
fundamental  series  can  be  compared  with  each  other 
to  determine  whether  they  are  equal  or  which  is  the 
greater  or  the  less  ;  they  thus  acquire  the  definiteness 
of  a  number  in  the  ordinary  sense.  A  number  defined 
by  a  fundamental  series  is  called  a  ''series  number." 
A  series  number  is  either  identical  with  a  rational 
number,  or  not  identical ;  in  the  latter  case  it  defines 
an  irrational  number.  The  domain  of  series  numbers 
consists  of  the  totality  of  all  rational  and  irrational 
numbers,  that  is  to  say,  of  all  real  numbers,  and  of 
these  only.  In  this  case  the  domain  of  real  numbers 
can  be  associated  with  a  straight  line,  as  G.  Cantor 
has  shown. 

The  extension  of  the  number-domain  by  the  addi- 
tion of  imaginary  quantities  is  closely  connected  with 
the  solution  of  equations,  especially  those  of  the  third 
degree.  The  Italian  algebraists  of  the  sixteenth  cen- 
tury called  them  "impossible  numbers."  As  proper 
solutions  of  an  equation,  imaginary  quantities  first 

*  Rosier,  Die  neueren  Definitionsformen  der  irrationalen  Zahlen,  1886. 


124  HISTORY  OF  MATHEMATICS. 

appear  in  the  writings  of  Albert  Girard*  (1629).  The 
expressions  "real"  and  "imaginary"  as  characteristic 
terms  for  the  difference  in  nature  of  the  roots  of  an 
equation  are  due  to  Descartes. f  De  Moivre  and  Lam- 
bert introduced  imaginary  quantities  into  trigonom- 
etry, the  former  by  means  of  his  famous  proposition 
concerning  the  power  (cos  <f>  +  /sin  <£)",  first  given  in 
its  present  form  by  Euler.  J 

Gauss§  added  to  his  great  fame  by  explaining  the 
nature  of  imaginary  quantities.  He  brought  into  gen- 
eral use  the  sign  /for  J/ — 1  first  suggested  by  Euler  :  |i 
he  calls  a-\-bi  a  complex  number  with  the  norm 
a2-|-<£2.  The  term  ' '  modulus  "  for  the  quantity  I/a2  -f-  //- 
conies  from  Argand  (1814),  the  term  "reduced  form" 
for  r(cos<£-|-  /sin  <£),  which  equals  a-\-btt  is  due  to 
Cauchy,  and  the  name  "direction  coefficient"  for  the 
factor  cos  <f>  -f-  /sin  <f>  first  appeared  in  print  in  an  essay 
of  Hankel's  (1861),  although  it  was  in  use  somewhat 
earlier.  Gauss,  to  whom  in  1799  it  seemed  simply 
advisable  to  retain  complex  numbers,^"  by  his  expla- 
nations in  the  advertisement  to  the  second  treatise  on 
biquadratic  residues  gained  for  them  a  triumphant 
introduction  into  arithmetic  operations. 

The  way  for  the  geometric  representation  of  com 
plex  quantities  was  prepared  by  the   observations  of 

*  Cantor,  II.,  p.  718.          t  Cantor,  II.,  p.  724.          J  Cantor,  HI.,  p.  68;. 
§  Hankel,  Die  komplexen  Zahh-n,  1867,  p.  71. 

|  Beman.  "Euler's  Use  of  i  to  Represent  an   Imaginary,"  Bull  A»i,->- 
Math.  Sac.,  March,  1898,  p.  274. 
^Treutlein. 


ALGEBRA.  125 

various  mathematicians  of  the  seventeenth  and  eight- 
eenth centuries,  among  them  especially  Wallis,*  who 
in  solving  geometric  problems  algebraically  became 
aware  of  the  fact  that  when  certain  assumptions  give 
two  real  solutions  to  a  problem  as  points  of  a  straight 
line,  other  assumptions  give  two  "impossible"  roots 
as  the  points  of  a  straight  line  perpendicular  to  the  first 
one.  The  first  satisfactory  representation  of  complex 
quantities  in  a  plane  was  devised  by  Caspar  Wessel 
in  1797,  without  attracting  the  attention  it  deserved. 
A  similar  treatment,  but  wholly  independent,  was  given 
by  Argand  in  1806.f  But  his  publication  was  not  ap- 
preciated even  in  France.  In  the  year  1813  there  ap- 
peared in  Gergonne's  Annales  by  an  artillery  officer 
Francais  in  the  city  of  Metz  the  outlines  of  a  theory 
of  imaginary  quantities  the  main  ideas  of  which  can 
be  traced  back  to  Argand.  Although  Argand  im- 
proved his  theory  by  his  later  work,  yet  it  did  not 
gain  recognition  until  Cauchy  entered  the  lists  as  its 
champion.  It  was,  however,  Gauss  who  (1831),  by 
means  of  his  great  reputation,  made  the  representa- 
tion of  imaginary  quantities  in  the  "Gaussian  plane" 
the  common  property  of  all  mathematicians.  J 

Gauss  and  Dirichlet  introduced  general  complex 
numbers  into  arithmetic.  The  primary  investigations 

*  Hankel,  Die  komplexen  Zaklen,  1867,  p.  81. 

t  Hankel,  Die  komplexen  Zahlen,  1867,  p.  82. 

JFor  a  rfsumf  of  the  history  of  the  geometric  representation  of  the  im- 
aginary, see  Beman,  "A  Chapter  in  the  History  of  Mathematics,"  Proc. 
Amer.  Assn.  Adv.  Science,  1897,  pp.  33-50 


I  26  HISTORY  OF  MATHEMATICS. 

of  Dirichlet  in  regard  to  complex  numbers,  which,  to- 
gether with  indications  of  the  proof,  are  contained  in 
the  Berichte  der  Berliner  Akademie  for  1841,  1842,  and 
1846,  received  material  amplifications  through  Eisen- 
stein,  Kummer,  and  Dedekind.  Gauss,  in  the  devel- 
opment of  the  real  theory  of  biquadratic  residues, 
introduced  complex  numbers  of  the  form  a  +  bi,  and 
Lejeune  Dirichlet  introduced  into  the  new  theory  of 
complex  numbers  the  notions  of  prime  numbers, 
congruences,  residue-theorems,  reciprocity,  etc  ,  the 
propositions,  however,  showing  greater  complexity 
and  variety  and  offering  greater  difficulties  in  the  way 
of  proof.*  Instead  of  the  equation  x*  —  1  =  0,  which 
gives  as  roots  the  Gaussian  units,  +  1,  —  1,  +  /,  —  /, 
Eisenstein  made  use  of  the  equation  x*  —  1  =  0  and 
considered  the  complex  numbers  a  +  bp  (p  being  a 
complex  cube  root  of  unity)  the  theory  resembling  that 
of  the  Gaussian  numbers  a-\-bi,  but  yet  possessing 
certain  marked  differences.  Kummer  generalized  the 
theory  still  further,  using  the  equation  x"  —  1  =  0  as 
the  basis,  so  that  numbers  of  the  form 


arise  where  the  a's  are  real  integers  and  the  A's  are 
roots  of  the  equation  x"  —  1  =  0.  Kummer  also  set 
forth  the  concept  of  ideal  numbers,  that  is,  of  such 
numbers  as  are  factors  of  prime  numbers  and  possess 
the  property  that  there  is  always  a  power  of  these  ideal 
numbers  which  gives  a  real  number.  For  example, 

*C«yley,  Address  to  the  British  Association,  etc.,  1883. 


ALGEBRA.  12J 

there  exists  for  the  prime  number/  no  rational  factor- 
ization so  that/3—  A-B  (where  A  is  different  from  / 
and  /2);  but  in  the  theory  of  numbers  formed  from 
the  twenty-third  roots  of  unity  there  are  prime  num- 
bers /  which  satisfy  the  condition  named  above.  In 
this  case  /  is  the  product  of  two  ideal  numbers,  of 
which  the  third  powers  are  the  real  numbers  A  and 
S,  so  that/3  =  ^-^.  In  the  later  development  given 
by  Dedekind  the  units  are  the  roots  of  any  irreducible 
equation  with  integral  numerical  coefficients.  In  the 
case  of  the  equation  x*— x  +  1=0,  £(l  +  /i/3),  that 
is  to  say,  the  p  of  Eisenstein,  is  to  be  regarded  as  in- 
tegral. 

In  tracing  out  the  nature  of  complex  numbers, 
H.  Grassmann,  Hamilton,  and  Scheffler  have  arrived 
at  peculiar  discoveries.  Grassmann,  who  also  mate- 
rially developed  the  theory  of  determinants,  investi- 
gated in  his  treatise  on  directional  calculus  (Ausdeh- 
nungslehre)  the  addition  and  multiplication  of  complex 
numbers.  In  like  manner,  Hamilton  originated  the 
calculus  of  quaternions,  a  method  of  calculation  re- 
garded with  especial  favor  in  England  and  America 
and  justified  by  its  relatively  simple  applicability  to 
spherics,  to  the  theory  of  curvature,  and  to  mechanics. 

The  complete  double  title*  of  H.  Grassmann's 
chief  work  which  appeared  in  the  year  1844,  as 
translated,  is:  "The  Science  of  Extensive  Quantities 
or  Directional  Calculus  (Ausdchnungslehre).  A  New 

*V.  Schlegel,  Grassmann,  sein  Lebeii  und  seine  Werke. 


128  HISTORY  OF  MATHEMATICS. 

Mathematical  Theory,  Set  Forth  and  Elucidated  by 
Applications.  Part  First,  Containing  the  Theory  of 
Lineal  Directional  Calculus.  The  Theory  of  Lineal 
Directional  Calculus,  A  New  Branch  of  Mathematics, 
Set  Forth  and  Elucidated  by  Applications  to  the 
Remaining  Branches  of  Mathematics,  as  well  as  to 
Statics,  Mechanics,  the  Theory  of  Magnetism  and 
Crystallography."  The  favorable  criticisms  of  this 
wonderful  work  by  Gauss,  who  discovered  that  "the 
tendencies  of  the  book  partly  coincided  with  the  paths 
upon  which  he  had  himself  been  travelling  for  half  a 
century,"  by  Grunert,  and  by  Mobius  who  recognised 
in  Grassmann  "a  congenial  spirit  with  respect  to 
mathematics,  though  not  to  philosophy,"  and  who 
congratulated  Grassmann  upon  his  "excellent  work," 
were  not  able  to  secure  for  it  a  large  circle  of  readers. 
As  late  as  1853  Mobius  stated  that  "Bretschneider 
was  the  only  mathematician  in  Gotha  who  had  assured 
him  that  he  had  read  the  Ausdehnungslehre  through.1' 
Grassmann  received  the  suggestion  for  his  re- 
searches from  geometry,  where  At  B,  C,  being  points 
of  a  straight  line,  A£  +  £C=AC.*  With  this  he 
combined  the  propositions  which  regard  the  parallel© 
gram  as  the  product  of  two  adjacent  sides,  thus  intro 
ducing  new  products  for  which  the  ordinary  rules  of 
multiplication  hold  so  long  as  there  is  no  permutation 
of  factors,  this  latter  case  requiring  the  change  of 


*  Grassmann,  Die  Ausdehnungslehre  von  1844  oder  die  lineale  Avsdeh- 
ngslehre,  ein  nruer  Ziveig  tier  Matltematik.     Zweite  Auflage,  1878. 


ALGEBRA.  12  Q 

signs.  More  exhaustive  researches  led  Grassmann  to 
regard  as  the  sum  of  several  points  their  center  of 
gravity,  as  the  product  of  two  points  the  finite  line- 
segment  between  them,  as  the  product  of  three  points 
the  area  of  their  triangle,  and  as  the  product  of  four 
points  the  volume  of  their  pyramid.  Through  the 
study  of  the  Barycentrischer  Calciil  of  Mobius,  Grass- 
mann was  led  still  further.  The  product  of  two  line- 
segments  which  form  a  parallelogram  was  called  the 
"external  product"  (the  factors  can  be  permuted  only 
by  a  change  of  sign),  the  product  of  one  line-segment 
and  the  perpendicular  projection  of  another  upon  it 
formed  the  "internal  product"  (the  factors  can  here 
be  permuted  without  change  of  sign).  The  introduc- 
tion of  the  exponential  quantity  led  to  the  enlarge- 
ment of  the  system,  of  which  Grassmann  permitted  a 
brief  survey  to  appear  in  Grunerfs  Archiv  (1845).* 

Hamiltonf  gave  for  the  first  time,  in  a  communi- 
cation to  the  Academy  of  Dublin  in  1844,  the  values 
/',  j,  k,  so  characteristic  of  his  theory.  The  Lectures 
on  Quaternions  appeared  in  1853,  the  Elements  of  Qua- 
ternions in  1866.  From  a  fixed  point  O  let  a  linej  be 
drawn  to  the  point  P  having  the  rectangular  co-ordi- 
nates x,  y,  z.  Now  if  /,  j,  k  represent  fixed  coefficients 
(unit  distances  on  the  axes),  then 

*  Translated  by  Beman,  Analyst,  1881,  pp.  96,  114. 

t  Unverzagt,  Theorie  der  goniotnetrischcn  und  longimetrischen  Quater- 
nionen,  1876. 

*Cayley,  A.,  "On  Multiple  Algebra,"  in  Quarterly  Journal  of  Mathe- 
matics, 1887. 


130  HISTORY  OF  MATHEMATICS. 


is  a  vector,  and  this  additively  joined  to  the  "pure 
quantity"  or  "scalar"  w  produces  the  quaternion 


The   addition  of   two   quaternions  follows   from    the 
usual  formula 


But  in  the  case  of  multiplication  we  must  place 

i*  ==/»  =  &=—!,  i  =jk  =  —  kj,  j  =  ki  =  —  ik, 

k  =  ij=—ji, 
so  that  we  obtain 
Q-Q'  =  ww'  —  xx'  —yy'  —  zz' 

-\-  i(wx'  -\-  xw'  -\-  yz'  —  z/) 
+  J(i*>y  +  yu/  +  *x'  —  xz) 
-\-k(wz'  -\-  zw'  -{-  xy'  —  yx'\ 

On  this  same  subject  Scheffler  published  in  184G 
his  first  work,  Ueber  die  Verhdltnisse  der  Arithmetik  zur 
Geometric,  in  1852  the  Situationscalcul,  and  in  1880  the 
Polydimensionalen  Grossen.  For  him  *  the  vector  r  in 
three  dimensions  is  represented  by 

r  =  a-e*y=T'SyTi,  or 
r  =  x  +y  V—  T  +  z  V  —  1  •  1/TT,  or 
r=x-\-yi-\-Z'i'i\    where   i=|X  —  1    and   /j^i/n-l 
are  turning  factors  of  an  angle  of  90°  in  the  plane  of  xy 
and  xz.   In  Scheffler's  theory  the  distributive  law  does 
not  always  hold  true  for  multiplication,  that  is  to  say, 
a(b-\-c)  is  not  always  equivalent  to  ab-\-  ac. 

Investigations  as  to  the  extent  of  the  domain  in 

*Unverzagt,  Ueber  die  Grundlagen  der  Kfchnung  tnit  Quaterniontn,  1881. 


which  with  certain  assumptions  the  laws  of  the  ele- 
mentary operations  of  arithmetic  are  valid  have  led 
to  the  establishment  of  a  calculus  of  logic.*  To  this 
class  of  investigations  there  belong,  besides  Grass- 
mann's  Formenlehre  (1872),  notes  by  Cayley  and  Ellis, 
and  in  particular  the  works  of  Boole,  SchrOder,  and 
Charles  Peirce. 

A  minor  portion  of  the  modern  theory  of  numbers 
or  higher  arithmetic,  which  concerns  the  theories  of 
congruences  and  of  forms,  is  made  up  of  continued 
fractions.  The  algorism  leading  to  the  formation  of 
such  fractions,  which  is  also  used  in  calculating  the 
greatest  common  measure  of  two  numbers,  reaches 
back  to  the  time  of  Euclid.  The  combination  of  the 
partial  quotients  in  a  continued  fraction  originated 
with  Cataldi,f  who  in  the  year  1613  approximated  the 
value  of  square  roots  by  this  method,  but  failed  to 
examine  closely  the  properties  of  the  new  fractions. 

Daniel  Schwenter  was  the  first  to  make  any  ma- 
terial contribution  (1625)  towards  determining  the 
convergents  of  continued  fractions.  He  devoted  his 
attention  to  the  reduction  of  fractions  involving  large 
numbers,  and  determined  the  rules  now  in  use  for  cal- 
culating the  successive  convergents.  Huygens  and 
Wallis  also  labored  in  this  field,  the  latter  discover- 
ing the  general  rule,  together  with  a  demonstration, 
.vhich  combines  the  terms  of  the  convergents 

*  SchrSder,  Der  Operationskreis  des  Logikcalculs,  1877. 
t  Cantor,  II.,  p.  695. 


133  HISTORY  OF  MATHEMATICS. 

fn        A-l        A-2 
?«'       ?— 1*       ?—S 

in  the  following  manner : 

A  _  "«P~-\  +  ^.A-a 

4n  an  <?»-!  +  &n  Qn-a 

The  theory  of  continued  fractions  received  its  greatest 
development  in  the  eighteenth  century  with  Euler,* 
who  introduced  the  name  fractio  continua  (the  Ger- 
man term  Kettenbruch  has  been  used  only  since  the 
beginning  of  the  nineteenth  century).  He  devoted 
his  attention  chiefly  to  the  reduction  of  continued 
fractions  to  the  form  of  infinite  products  and  series, 
and  doubtless  in  this  way  was  led  to  the  attempt  to 
give  the  convergents  independent  form,  that  is  to  dis- 
cover a  general  law  by  means  of  which  it  would  be 
possible  to  calculate  any  required  convergent  without 
first  obtaining  the  preceding  ones.  Although  Euler 
did  not  succeed  in  discovering  such  a  law,  he  created 
an  algorism  of  some  value.  This,  however,  did  not 
bring  him  essentially  nearer  the  goal  because,  in  spite 
of  the  example  of  Cramer,  he  neglected  to  make  use 
of  determinants  and  thus  to  identify  himself  the  more 
closely  with  the  pure  theory  of  combinations.  From 
this  latter  point  of  view  the  problem  was  attacked  by 
Hindenburg  and  his  pupils  Burckhardt  and  Rothe. 
Still,  those  who  proceed  from  the  theory  of  combina 
tions  alone  know  continued  fractions  only  from  one 
side;  the  method  of  independent  presentation  allows 

*  Cantor,  III.,  p.  670. 


ALGEBRA.  133 

the  calculation  of  the  desired  convergent  from  both 
sides,  forward  as  well  as  backward,  to  the  practical 
value  of  which  Dirichlet  has  testified. 

Only  in  modern  times  has  the  calculus  of  determi- 
nants been  employed  in  this  field,  together  with  a 
combinatory  symbol,  and  the  first  impulse  in  this  di- 
rection dates  from  the  Danish  mathematician  Ramus 
(1855).  Similar  investigations  were  begun,  however, 
by  Heine,  MObius,  and  S.  Gunther,  leading  to  the 
formation  of  "continued  fractional  determinants.'1 
The  irrationality  of  certain  infinite  continued  frac- 
tions* had  been  investigated  before  this  by  Legendre, 
who,  like  Gauss,  gave  the  quotient  of  two  power  se- 
ries in  the  form  of  a  continued  fraction.  By  means  of 
the  application  of  continued  fractions  it  can  be  shown 
that  the  quantities  c*  (for  rational  values  of  x),  e,  », 
and  ir2  cannot  be  rational  (Lambert,  Legendre,  Stern). 
It  was  not  until  very  recent  times  that  the  transcen- 
dental nature  of  e  was  established  by  Hermite,  and 
that  of  it  by  F.  Lindemann. 

In  the  theory  of  numbers  strictly  speaking,  quite 
difficult  problems  concerning  the  properties  of  num- 
bers were  solved  by  the  first  exponents  of  that  study, 
Euclid  and  Diophantus.  Any  considerable  advance 
was  impossible,  however,  as  long  as  investigations  had 
to  be  conducted  f  without  an  adequate  numerical  nota- 
tion, and  almost  exclusively  with  the  aid  of  an  algebra 

*Treutlein. 

+  Legendre,  Thtorie  des  nombres,  ist  ed.  1798,  3rd  ed.  1830. 


134  HISTORY  OF  MATHEMATICS. 

just  developing  under  the  guise  of  geometry.  Until 
the  time  of  Vieta  and  Bachet  there  is  no  essential  ad- 
vance to  be  noted  in  the  theory  of  numbers.  The 
former  solved  many  problems  in  this  field,  and  the 
latter  gave  in  his  work  Problemes  plaisants  et  dtlectables 
a  satisfactory  treatment  of  indeterminate  equations 
of  the  first  degree.  Still  later  the  first  stones  for  the 
foundation  of  a  theory  of  numbers  were  laid  by  Fer- 
mat,  who  had  carefully  studied  Diophantus  and  into 
whose  works  as  elaborated  by  Bachet  he  incorporated 
valuable  additional  propositions.  The  great  mass  of 
propositions  which  can  be  traced  back  to  him  he  gave 
for  the  most  part  without  demonstration,  as  for  ex- 
ample the  following  statement : 

"Every  prime  number  of  the  form  4»-j-l  is  the 
sum  of  two  squares;  a  prime  number  of  the  form 
8«-fl  has  at  the  same  time  the  three  forms  j'-f-z2, 
y*-\-2z*,  y*  —  2z* ;  every  prime  number  of  the  form 
Sn  +  3  appears  as  y*  -f-  2s3,  every  one  of  the  form  8«-f  7 
appears  as  y1  —  2zV  Further,  "Any  number  can  be 
formed  by  the  addition  of  three  cubes,  of  four  squares, 
of  five  fifth  powers,  etc." 

Fermat  proved  that  the  area  of  a  Pythagorean 
right-angled  triangle,  for  example  a  triangle  with  the 
sides  3,  4,  and  5,  cannot  be  a  square.  He  was  also 
the  first  to  obtain  the  solution  of  the  equation  ax*  -f 
1  =y*,  where  a  is  not  a  square ;  at  all  events,  he 
brought  this  problem  to  the  attention  of  English 
mathematicians,  among  whom  Lord  Brouncker  dis- 


ALGEBRA.  135 

covered  a  solution  which  found  its  way  into  the 
works  of  Wallis.  Many  of  Fermat's  theorems  belong 
to  "the  finest  propositions  of  higher  mathematics,"* 
and  possess  the  peculiarity  that  they  can  easily  be 
discovered  by  induction,  but  that  their  demonstrations 
are  extremely  difficult  and  yield  only  to  the  most 
searching  investigation.  It  is  just  this  which  imparts 
to  higher  arithmetic  that  magic  charm  which  made  it 
a  favorite  with  the  early  geometers,  not  to  speak  of 
its  inexhaustible  treasure-house  in  which  it  far  ex- 
ceeds all  other  branches  of  pure  mathematics. 

After  Fermat,  Euler  was  the  first  again  to  attempt 
any  serious  investigations  in  the  theory  of  numbers. 
To  him  we  owe,  among  other  things,  the  first  scien- 
tific solution  of  the  chess  board  problem,  which  re- 
quires that  the  knight,  starting  from  a  certain  square, 
shall  in  turn  occupy  all  sixty-four  squares,  and  the 
further  proposition  that  the  sum  of  four  squares  mul- 
tiplied into  another  similar  sum  also  gives  the  sum  of 
four  squares.  He  also  discovered  demonstrations  of 
various  propositions  of  Fermat,  as  well  as  the  general 
solution  of  indeterminate  equations  of  the  second  de- 
gree with  two  unknowns  on  the  hypothesis  that  a  spe- 
cial solution  is  known,  and  he  treated  a  large  number 
of  other  indeterminate  equations,  for  which  he  dis- 
covered numerous  ingenious  solutions. 

Euler  (as  well  as  Krafft)  also  occupied  himself 

*  Gauss,  Werke,  II.,  p.  152. 


136  HISTORY  OF  MATHEMATICS. 

with  amicable  numbers.*  These  numbers,  which  are 
mentioned  by  lamblichus  as  being  known  to  the 
Pythagoreans,  and  which  are  mentioned  by  the  Arab 
Tabit  ibn  Kurra,  suggested  to  Descartes  the  discovery 
of  a  law  of  formation,  which  is  given  again  by  Van 
Schooten.  Euler  made  additions  to  this  law  and  de- 
duced from  it  the  proposition  that  two  amicable  num 
bers  must  possess  the  same  number  of  prime  factors. 
The  formation  of  amicable  numbers  depends  either 
upon  the  solution  of  the  equation  xy  -f  ax  -f-  by  -\-  c  =  0, 
or  upon  the  factoring  of  the  quadratic  form  ax*  -f-  bxy 


Following  Euler,  Lagrange  was  able  to  publish 
many  interesting  results  in  the  theory  of  numbers. 
He  showed  that  any  number  can  be  represented  as 
the  sum  of  four  or  less  squares,  and  that  a  real  root 
of  an  algebraic  equation  of  any  degree  can  be  con- 
verted into  a  continued  fraction.  He  was  also  the 
first  to  prove  that  the  equation  x*  —  Ay*  =  \  is  always 
soluble  in  integers,  and  he  discovered  a  general  method 
for  the  derivation  of  propositions  concerning  prime 
numbers. 

Now  the  development  of  the  theory  of  numbers 
bounds  forward  in  two  mighty  leaps  to  Legendre  and 
Gauss.  The  valuable  treatise  of  the  former,  Essai  sur 
la  thtorie  des  nombres,  which  appeared  but  a  few  years 
before  Gauss's  Disquisitiones  arithmeticae,  contains  an 
epitome  of  all  results  that  had  been  published  up  to 

*Seelhoff,  "  Befreundete  Zahlen,"  Hoppe  Arch.,  Bd.  70. 


ALGEBRA.  137 

that  time,  besides  certain  original  investigations,  the 
most  brilliant  being  the  law  of  quadratic  reciprocity, 
or,  as  Gauss  called  it,  the  Theorema  fundamentale  in 
doctrina  de  rcsiduis  quadratis.  This  law  gives  a  rela- 
tionship between  two  odd  and  unequal  prime  numbers 
and  can  be  enunciated  in  the  following  words : 

"Let  ( — )  be  the  remainder  which  is  left  after  divid- 
\*J 


ing  m  **     by  n,  and  let    —  )  be  the  remainder  left  after 

>*-i  \mJ 

dividing   n  2     by  m.     These  remainders   are  always 

-f-  1  or  —  1.     Whatever  then  the  prime  numbers  m 

and  n  may  be,  we  always  obtain  (—)  =  (  —  )  in  case  the 

\mj       \*j 
numbers  are  not  both  of  the  form  4*  -\-  3.    But  if  both 

are  of  the  form  4*+  3,  then  we  have  f—  J  =  —  (—  )•" 
These  two  cases  are  contained  in  the  formula 


, 

Bachet  having  exhausted  the  theory  of  the  indetermi- 
nate equation  of  the  first  degree  with  two  unknowns, 
an  equation  which  in  Gauss's  notation  appears  in  the 
form  x  =  a  (mod  £),  identical  with  -j-  =y-\-a,  mathe- 
maticians began  the  study  of  the  congruence  x*  =  m 
(mod  n).  Fermat  was  aware  of  a  few  special  cases  of 
the  complete  solution  ;  he  knew  under  what  conditions 
±  1,  2,  ±3,  5  are  quadratic  residues  or  non-residues 
of  the  odd  prime  number  »/.*  For  the  cases  —  1  and 

*Baumgart,   "  Ueber  das  quadratische   Reciprocitatsgesetz,"   in  SchlS- 
milch'tZtitschrifl,  Bd.  30,  HI.  Abt. 


138  HISTORY  OF  MATHEMATICS. 

±  3  the  demonstrations  originate  with  Euler,  for  ±  2 
and  ±  5  with  Lagrange.  It  was  Euler,  too,  who  gave 
the  propositions  which  embrace  the  law  of  quadratic 
reciprocity  in  the  most  general  terms,  although  he 
did  not  offer  a  complete  demonstration  of  it.  The 
famous  demonstration  of  Legendre  (in  Essai  sur  la 
theorie  des  nombres,  1798)  is  also,  as  yet,  incomplete. 
In  the  year  1796  Gauss  submitted,  without  knowing 
of  Euler's  work,  the  first  unquestionable  demonstra- 
tion— a  demonstration  which  possesses  at  the  same 
time  the  peculiarity  that  it  embraces  the  principles 
which  were  used  later.  In  the  course  of  time  Gauss 
adduced  no  less  than  eight  proofs  for  this  important 
law,  of  which  the  sixth  (chronologically  the  last)  was 
simplified  almost  simultaneously  by  Cauchy,  Jacobi. 
and  Eisenstein.  Eisenstein  demonstrated  in  partic 
ular  that  the  quadratic,  the  cubic  and  the  biquadratic 
laws  are  all  derived  from  a  common  source.  In  the 
year  1861  Kummer  worked  out  with  the  aid  of  the 
theory  of  forms  two  demonstrations  for  the  law  of 
quadratic  reciprocity,  which  were  capable  of  gene- 
ralization for  the  #th-power  residue.  Up  to  1890 
twenty-five  distinct  demonstrations  of  the  law  of 
quadratic  reciprocity  had  been  published ;  they  make 
use  of  induction  and  reduction,  of  the  partition  of  the 
perigon,  of  the  theory  of  functions,  and  of  the  theory 
of  forms.  In  addition  to  the  eight  demonstrations  by 
Gauss  which  have  already  been  mentioned,  there  are 
four  by  Eisenstein,  two  by  Kummer,  and  one  each 


ALGEBRA.  139 

by  Jacobi,  Cauchy,  Liouville,  Lebesgue,  Genocchi, 
Stern,  Zeller,  Kronecker,  Bouniakowsky,  Schering, 
Petersen,  Voigt,  Busche,  and  Pepin. 

However  much  is  due  to  the  co-operation  of  math- 
ematicians of  different  periods,  yet  to  Gauss  unques- 
tionably belongs  the  merit  of  having  contributed  in 
his  Disquisitiones  arithmeticae  of  1801  the  most  impor- 
tant part  of  the  elementary  development  of  the  theory 
of  numbers.  Later  investigations  in  this  branch  have 
their  root  in  the  soil  which  Gauss  prepared.  Of  such 
investigations,  which  were  not  pursued  until  after  the 
introduction  of  the  theory  of  elliptic  transcendents, 
may  be  mentioned  the  propositions  of  Jacobi  in  regard 
to  the  number  of  decompositions  of  a  number  into 
two,  four,  six,  and  eight  squares,*  as  well  as  the  in- 
vestigations of  Dirichlet  in  regard  to  the  equation 


His  work  in  the  theory  of  numbers  was  Dirichlet's 
favorite  pursuit,  f  He  was  the  first  to  deliver  lectures 
on  the  theory  of  numbers  in  a  German  university  and 
was  able  to  boast  of  having  made  the  Disquisitiones 
arithmeticae  of  Gauss  transparent  and  intelligible  —  a 
task  in  which  a  Legendre,  according  to  his  own 
avowal,  was  unsuccessful. 

Dirichlet's  earliest  treatise,  Mtmoire  sur  rimpossi- 
bilitc'  de  quelques  equations  indtt  ermine's  du  cinquieme 
degrc"  (submitted  to  the  French  Academy  in  1825), 

*  Dirichlet,  "  G«dachtnisrede  auf  Jacobi,"  Crellt't  Journal,  Bd.  52. 

t  Kummer,  "  Gedachtnisrede  auf  Lejeune-Dirichlet,"  in  Berl.  Abh.    1860. 


140  HISTORY  OF  MATHEMATICS. 

deals  with  the  proposition,  stated  by  Fermat  without 
demonstration,  that  "the  sum  of  two  powers  having 
the  same  exponent  can  never  be  equal  to  a  power  of 
the  same  exponent,  when  these  powers  are  of  a  degree 
higher  than  the  second."  Euler  and  Legendre  had 
proved  this  proposition  for  the  third  and  fourth  pow- 
ers ;  Dirichlet  discusses  the  sum  of  two  fifth  powers 
and  proves  that  for  integral  numbers  x5  -}-y&  cannot 
be  equal  to  az6.  The  importance  of  this  work  lies  in 
its  intimate  relationship  to  the  theory  of  forms  of 
higher  degree.  Dirichlet's  further  contributions  in  the 
field  of  the  theory  of  numbers  contain  elegant  demon- 
strations of  certain  propositions  of  Gauss  in  regard 
to  biquadratic  residues  and  the  law  of  reciprocity, 
which  were  published  in  1825  in  the  Gottingen  Ge- 
lehrte  Anzeigen,  as  well  as  with  the  determination  of 
the  class-number  of  the  quadratic  form  for  any  given 
determinant.  His  "applications  of  analysis  to  the 
theory  of  numbers  are  as  noteworthy  in  their  way  as 
Descartes's  applications  of  analysis  to  geometry.  They 
would  also,  like  the  analytic  geometry,  be  recognized 
as  a  new  mathematical  discipline  if  they  had  been  ex- 
tended not  to  certain  portions  only  of  the  theory  of 
number,  but  to  all  its  problems  uniformly.* 

The  numerous  investigations  into  the  properties 
and  laws  of  numbers  had  led  in  the  seventeenth  cen 
turyt  to  the  study  of  numbers  in  regard  to  their  divis- 

*Knmmer,  "Gedachtnisrede  auf  Lejeune-Dirichlet."    Berl.  Abh.     1860. 
tSeelhoff,  "Geschichte  der  Faktorentafeln,"  in  Hoppe  Arch.,  Bd.  70. 


141 


ors.  For  almost  two  thousand  years  Eratosthenes's 
"sieve"  remained  the  only  method  of  determining 
prime  numbers.  In  the  year  1657  Franz  van  Schooten 
published  a  table  of  prime  numbers  up  to  ten  thou- 
sand. Eleven  years  later  Pell  constructed  a  table  of 
the  least  prime  factors  (with  the  exception  of  2  and  5) 
of  all  numbers  up  to  100000.  In  Germany  these 
tables  remained  almost  unknown,  and  in  the  year 
1728  Poetius  published  independently  a  table  of  fac- 
tors for  numbers  up  to  100  000,  an  example  which 
was  repeatedly  imitated.  Kriiger's  table  of  1746  in- 
cludes numbers  up  to  100000;  that  of  Lambert  of 
1770,  which  is  the  first  to  show  the  arrangement 
used  in  more  modern  tables,  includes  numbers  up  to 
102000.  Of  the  six  tables  which  were  prepared  be- 
tween the  years  1770  and  1811  that  of  Felkel  is  inter- 
esting because  of  its  singular  fate ;  its  publication  by 
the  Kaiserlich  konigliches  Aerarium  in  Vienna  was 
completed  as  far  as  408  000 ;  the  remainder  of  the 
manuscript  was  then  withheld  and  the  portion  already 
printed  was  used  for  manufacturing  cartridges  for  the 
last  Turkish  war  of  the  eighteenth  century.  In  the 
year  1817  there  appeared  in  Paris  Burckhardt's  Table 
des  diviseurs  pour  tons  les  nombres  du  /"",  .2',  j*  million. 
Between  1840  and  1850  Crelle  communicated  to  the 
Berlin  Academy  tables  of  factors  for  the  fourth,  fifth, 
and  sixth  million,  which,  however,  were  not  pub- 
lished. Dase,  who  is  known  for  his  arithmetic  gen- 
ius, was  to  make  the  calculations  for  the  seventh  to 


142  HISTORY  OF  MATHEMATICS. 

the  tenth  million,  having  been  designated  for  that 
work  by  Gauss,  but  he  died  in  1861  before  its  com- 
pletion. Since  1877  the  British  Association  has  been 
having  these  tables  continued  by  Glaisher  with  the 
assistance  of  two  computers.  The  publication  of 
tables  of  factors  for  the  fourth  million  was  completed 
in  1879. 

In  the  year  1856  K.  G.  Reuschle  published  his 
tables  for  use  in  the  theory  of  numbers,  having  been 
encouraged  to  undertake  the  work  by  his  correspond- 
ence with  Jacobi.  They  contain  the  resolution  of 
numbers  of  the  form  10*  —  1  into  prime  factors,  up  to 
«  =  242,  and  numerous  similar  results  for  numbers  of 
the  form  a"  —  1,  and  a  table  of  the  resolution  of  prime 
numbers  /  =  Qn  -f  1  into  the  forms 
and  4= 


as  they  occur  in  the  treatment  of  cubic  residues  and 
in  the  partition  of  the  perigon. 

Of  greatest  importance  for  the  advance  of  the  sci- 
ence of  algebra  as  well  as  that  of  geometry  was  the 
development  of  the  theories  of  symmetric  functions, 
of  elimination,  and  of  invariants  of  algebraic  forms, 
as  they  were  perfected  through  the  application  of  pro- 
jective  geometry  to  the  theory  of  equations.* 

The  first  formulas  for  calculating  symmetric  func- 
tions (sums  of  powers)  of  the  roots  of  an  algebraic 
equation  in  terms  of  its  coefficients  are  due  to  Newton. 

*A.  Brill.  Antrittsrecie  in  Tubingen,  1884.     Manuscript. 


ALGEBRA.  143 

Waring  also  worked  in  this  field  (1770)  and  developed 
a  theorem,  which  Gauss  independently  discovered 
(1816),  by  means  of  which  any  symmetric  function 
may  be  expressed  in  terms  of  the  elementary  sym- 
metric functions.  This  is  accomplished  directly  by  a 
method  devised  by  Cayley  and  Sylvester,  through  laws 
due  to  the  former  in  regard  to  the  weight  of  sym 
metric  functions.  The  oldest  tables  of  symmetric 
functions  (extending  to  the  tenth  degree)  were  pub- 
lished by  Meyer-Hirsch  in  his  collection  of  problems 
(1809).  The  calculation  of  these  functions,  which  was 
very  tedious,  was  essentially  simplified  by  Cayley  and 
Brioschi. 

The  resultant  of  two  equations  with  one  unknown, 
or,  what  is  the  same,  of  two  forms  with  two  homo- 
geneous variables,  was  given  by  Euler  (1748)  and  by 
Bezout  (1764).  To  both  belongs  the  merit  of  having 
reduced  the  determination  of  the  resultant  to  the  so- 
lution of  a  system  of  linear  equations.*  Bezout  intro- 
duced the  name  "resultant"  (De  Morgan  suggested 
"eliminant")  and  determined  the  degree  of  this  func- 
tion. Lagrange  and  Poisson  also  investigated  ques- 
tions of  elimination ;  the  former  stated  the  condition 
for  common  multiple-roots;  the  latter  furnished  a 
method  of  forming  symmetric  functions  of  the  com- 
mon values  of  the  roots  of  a  system  of  equations.  The 
further  advancement  of  the  theory  of  elimination  was 
made  by  Jacobi,  Hesse,  Sylvester,  Cayley,  Cauchy, 

*  Salmon,  Higher  Algtbra. 


144  HISTORY  OF  MATHEMATICS. 

Brioschi,  and  Gordan.  Jacobi's  memoir,*  which  rep- 
resented the  resultant  as  a  determinant,  threw  light 
at  the  same  time  on  the  aggregate  of  coefficients  be- 
longing to  the  resultant  and  on  the  equations  in  which 
the  resultant  and  its  product  by  another  partially  ar- 
bitrary function  are  represented  as  functions  of  the 
two  given  forms.  This  notion  of  Jacobi  gave  Hesse 
the  impulse  to  pursue  numerous  important  investiga- 
tions, especially  on  the  resultant  of  two  equations, 
which  he  again  developed  in  1843  after  Sylvester's 
dialytic  method  (1840);  then  in  1844,  "on  the  elimi- 
nation of  the  variables  from  three  algebraic  equations 
with  two  variables";  and  shortly  after  "on  the  points 
of  inflexion  of  plane  curves."  Hesse  placed  the  main 
value  of  these  investigations,  not  in  the  form  of  the 
final  equation,  but  in  the  insight  into  the  composition 
of  the  same  from  known  functions.  Thus  he  came 
upon  the  functional  determinant  of  three  quadratic 
prime  forms,  and  further  upon  the  determinant  of  the 
second  partial  differential  coefficients  of  the  cubic 
form,  and  upon  its  Hessian  determinant,  whose  geo- 
metric interpretation  furnished  the  interesting  result 
that  in  the  general  case  the  points  of  inflexion  of  a 
plane  curve  of  the  «th  order  are  given  by  its  complete 
intersection  with  a  curve  of  order  3(«  — 2).  This  re- 
sult was  previously  known  for  curves  of  the  third 
order,  having  been  discovered  by  Pliicker.  To  Hesse 
is  further  due  the  first  important  example  of  the  re- 

*O.  H.  Noether,  Schlomilc/t's  Zeitschrift,  Bd.  ao. 


ALGEBRA.  145 

moval  of  factors  from  resultants,  in  so  far  as  these 
factors  are  foreign  to  the  real  problem  to  be  solved. 
Hesse,  always  extending  the  theory  of  elimination, 
in  1849  succeeded  in  producing,  free  from  all  super- 
fluous factors,  the  long-sought  equation  of  the  14th 
degree  upon  which  the  double  tangents  of  a  curve  of 
the  4th  order  depend. 

The  method  of  elimination  used  by  Hesse*  in  1843 
is  the  dialytic  method  published  by  Sylvester  in  1840 ; 
it  gives  the  resultant  of  two  functions  of  the  mth  and 
nth  orders  as  a  determinant,  in  which  the  coefficients 
of  the  first  enter  into  n  rows,  and  those  of  the  second 
into  m  rows.  It  was  Sylvester  also,  who  in  1851  in- 
troduced the  name  "discriminant"  for  the  function 
which  expresses  the  condition  for  the  existence  of 
two  equal  roots  of  an  algebraic  equation  ;  up  to  this 
time,  it  was  customary,  after  the  example  of  Gauss, 
to  say  "determinant  of  the  function." 

The  notion  of  invariance,  so  important  for  all 
branches  of  mathematics  to-day,  dates  back  in  its 
beginnings  to  Lagrangef,  who  in  1773  remarked 
that  the  discriminant  of  the  quadratic  form  ax2  -(- 
Zbxy  -\-  cy1  remains  unaltered  by  the  substitution  of 
x-\-\y  for  x.  This  unchangeability  of  the  discrim- 
inant by  linear  transformation,  for  binar)'  and  ternary 
quadratic  forms,  was  completely  proved  by  Gauss 
(1801)  ;  but  that  the  discriminant  in  general  and  in 
every  case  remains  invariant  by  linear  transformation, 

*  Matthiessen,  p.  99.  t  Salmon,  Higher  Algebra. 


146  HISTORY  OF  MATHEMATICS. 

G.  Boole  (1841)  recognized  and  first  demonstrated. 
In  1845,  Cayley,  adding  to  the  treatment  of  Boole, 
found  that  there  are  still  other  functions  which  possess 
invariant  properties  in  linear  transformation,  showed 
how  to  determine  such  functions  and  named  them 
"hyperdeterminants."  This  discovery  of  Cayley  de- 
veloped rapidly  into  the  important  theory  of  invari- 
ants, particularly  through  the  writings  of  Cayley, 
Aronhold,  Boole,  Sylvester,  Hermite,  and  Brioschi, 
and  then  through  those  of  Clebsch,  Gordan,  and 
others.  After  the  appearance  of  Cayley's  first  paper, 
Aronhold  made  an  important  contribution  by  deter- 
mining the  invariants  S  and  T  of  a  ternary  form,  and 
by  developing  their  relation  to  the  discriminant  of 
the  same  form.  From  1851  on,  there  appeared  a  se- 
ries of  important  articles  by  Cayley  and  Sylvester. 
The  latter  created  in  these  a  large  part  of  the  termin 
ology  of  to-day,  especially  the  name  "invariant" 
(1851).  In  the  year  1854,  Hermite  discovered  his  law 
of  reciprocity,  which  states  that  to  every  covariant  or 
invariant  of  degree  p  and  order  r  of  a  form  of  the  mtli 
order,  corresponds  also  a  covariant  or  invariant  of 
degree  m  and  of  order  r  of  a  form  of  the  pth  order. 
Clebsch  and  Gordan  used  the  abbreviation  £",  intro- 
duced for  binary  forms  by  Aronhold,  in  their  funda- 
mental developments,  e.  g.,  in  the  systematic  ex- 
tension of  the  process  of  transvection  in  forming 
invariants  and  covariants,  already  known  to  Cayley 
in  his  preliminary  investigations,  in  the  folding-pro- 


ALGEBRA.  147 

cess  of  forming  elementary  covariants,  and  in  the  for- 
mation of  simultaneous  invariants  and  covariants,  in 
particular  the  combinants.  Gordan's  theorem  on  the 
finiteness  of  the  form-system  constitutes  the  most  im- 
portant recent  advance  in  this  theory ;  this  theorem 
states  that  there  is  only  a  finite  number  of  invariants 
and  covariants  of  a  binary  form  or  of  a  system  of  such 
forms.  Gordan  has  also  given  a  method  for  the  for- 
mation of  the  complete  form-system,  and  has  carried 
out  the  same  for  the  case  of  binary  forms  of  the  fifth 
and  sixth  orders.  Hilbert  (1890)  showed  the  finite- 
ness  of  the  complete  systems  for  forms  of  n  variables.* 
To  refer  in  a  word  to  the  great  significance  of  the  theory  of 
invariants  for  other  branches  of  mathematics,  let  it  suffice  to 
mention  that  the  theory  of  binary  forms  has  been  transferred  by 
Clebsch  to  that  of  ternary  forms  (in  particular  for  equations  in 
line  co-ordinates) ;  that  the  form  of  the  third  order  finds  its  repre- 
sentation in  a  space-curve  of  the  third  order,  while  binary  forms 
of  the  fourth  order  play  a  great  part  in  the  theory  of  plane  curves 
of  the  third  order,  and  assist  in  the  solution  of  the  equation  of 
the  fourth  degree  as  well  as  in  the  transformation  of  the  elliptic 
integral  of  the  first  class  into  Hermite's  normal  form ;  finally  that 
combinants  can  be  effectively  introduced  in  the  transformation  of 
equations  of  the  fifth  and  sixth  degrees.  The  results  of  investiga- 
tions by  Clebsch,  Weierstrass,  Klein,  Bianchi,  and  Burckhardt, 
have  shown  the  great  significance  of  the  theory  of  invariants  for 
the  theory  of  the  hyperelliptic  and  Abelian  functions.  This  theory 
has  been  further  used  by  Christoffel  and  Lipschitz  in  the  represen- 
tation of  the  line-element,  by  Sylvester,  Halphen,  and  Lie  in  the 
case  of  reciprocants  or  differential  invariants  in  the  theory  of  dif- 

*  Meyer,  W.  F.,  "  Bericht  iiber  den  gegenwartigen  Stand  der  Invarianten- 
theorie."    Jahresbericht  der  deutschen  Mathemaliker-Vereinigung,  Bd.  I. 


148  HISTORY  OF  MATHEMATICS. 

ferential  equations,  and  by  Beltrami  in  his  differential  parameter 
in  the  theory  of  curvature  of  surfaces.  Irrational  invariants  also 
have  been  proposed  in  various  articles  by  Hilbert. 

The  theory  of  probabilities  assumed  form  under 
the  hands  of  Pascal  and  Fermat*  In  the  year  1654, 
a  gambler,  the  Chevalier  de  Mere,  had  addressed  two 
inquiries  to  Pascal  as  follows :  "  In  how  many  throws 
with  dice  can  one  hope  to  throw  a  double  six,"  and 
"In  what  ratio  should  the  stakes  be  divided  if  the 
game  is  broken  up  at  a  given  moment?"  These  two 
questions,  whose  solution  was  for  Pascal  very  easy, 
were  the  occasion  of  his  laying  the  foundation  of  a 
new  science  which  was  named  by  him  "  Geometric  du 
hasard."  At  Pascal's  invitation,  Fermat  also  turned 
his  attention  to  such  questions,  using  the  theory  of 
combinations.  Huygens  soon  followed  the  example 
of  the  two  French  mathematicians,  and  wrote  in  1656f 
a  small  treatise  on  games  of  chance.  The  first  to 
apply  the  new  theory  to  economic  sciences  was  the 
"grand  pensioner"  Jean  de  Witt,  the  celebrated  pupil 
of  Descartes.  He  made  a  report  in  1671  on  the  man- 
ner of  determining  the  rate  of  annuities  on  the  basis 
of  a  table  of  mortality.  Hudde  also  published  in- 
vestigations on  the  same  subject.  "Calculation  of 
chances"  {Rechnung  iibcr  den  Zufall}  received  compre- 
hensive treatment  at  the  hand  of  Jacob  Bernoulli  in 
his  Ars  conjectandi  (1713),  printed  eight  years  after  the 
death  of  the  author,  a  book  which  remained  forgotten 

*  Cantor,  II.,  p.  688.  t  Cantor,  II.,  p.  692. 


ALGEBRA.  149 

until  Condorcet  called  attention  to  it.  Since  Ber- 
noulli, there  has  scarcely  been  a  distinguished  alge- 
braist who  has  not  found  time  for  some  work  in  the 
theory  of  probabilities. 

To  the  method  of  least  squares  Legendre  gave  the 
name  in  a  paper  on  this  subject  which  appeared  in 
1805.*  The  first  publication  by  Gauss  on  the  same 
subject  appeared  in  1809,  although  he  was  in  posses- 
sion of  the  method  as  early  as  1795.  The  honor  is 
therefore  due  to  Gauss  for  the  reason  that  he  first  set 
forth  the  method  in  its  present  form  and  turned  it  to 
practical  account  on  a  large  scale.  The  apparent  in- 
spiration for  this  investigation  was  the  discovery  of 
the  first  planetoid  Ceres  on  the  first  of  January,  1801, 
by  Piazzi.  Gauss  calculated  by  new  methods  the 
orbit  of  this  heavenly  body  so  accurately  that  the 
same  planetoid  could  be  again  found  towards  the  end 
of  the  year  1801  near  the  position  given  by  him.  The 
investigations  connected  with  this  calculation  ap- 
peared in  1809  as  Theoria  motus  corporum  coelestium, 
etc.  The  work  contained  the  determination  of  the 
position  of  a  heavenly  body  for  any  given  time  by 
means  of  the  known  orbit,  besides  the  solution  of  the 
difficult  problem  to  find  the  orbit  from  three  observa- 
tions. In  order  to  make  the  orbit  thus  determined 
agree  as  closely  as  possible  with  that  of  a  greater 
number  of  observations,  Gauss  applied  the  process 

*Merriman,   M.,   "List  of  Writings    relating  to  the   Method  of  Least 
Squares."     Trans.  Conn.  Acad.,  Vol.  IV. 


150  HISTORY  OF  MATHEMATICS. 

discovered  by  him  in  1795.  The  object  of  this  was 
"so  to  combine  observations  which  serve  the  purpose 
of  determining  unknown  quantities,  that  the  unavoid- 
able errors  of  observation  affect  as  little  as  possible 
the  values  of  the  numbers  sought."  For  this  purpose 
Gauss  gave  the  following  rule*:  "Attribute  to  each 
error  a  moment  depending  upon  its  value,  multiply 
the  moment  of  each  possible  error  by  its  probability 
and  add  the  products.  The  error  whose  moment  is 
equal  to  this  sum  will  have  to  be  designated  as  the 
mean."  As  the  simplest  arbitrary  function  of  the 
error  which  shall  be  the  moment  of  the  latter,  Gauss 
chose  the  square.  Laplace  published  in  the  year  1812 
a  detailed  proof  of  the  correctness  of  Gauss's  method. 
Elementary  presentations  of  the  theory  of  combi- 
nations are  found  in  the  sixteenth  century,  e.  g.,  by 
Cardan,  but  the  first  great  work  is  due  to  Pascal.  In 
this  he  uses  his  arithmetic  triangle,  in  order  to  de- 
termine the  number  of  combinations  of  m  elements  of 
the  nth  class.  Leibnitz  and  Jacob  Bernoulli  produced 
much  new  material  by  their  investigations.  Towards 
the  end  of  the  eighteenth  century,  the  field  was  cul- 
tivated by  a  number  of  German  scholars,  and  there 
arose  under  the  leadership  of  Hindenburg  the  "com- 
binatory  school,"f  whose  followers  added  to  the  de- 
velopment of  the  binomial  theorem.  Superior  to  them 
all  in  systematic  proof  is  Hindenburg,  who  separated 

*Gerhardt,  Geschichte  der  Mathematik  in  Deutschland,  1877. 
tGerhardt,  Geschichtt  tier  Mathematik  in  Deutschland,  1877. 


polynomials  into  a  first  class  of  the  form  a  -\-  b -f-  c  -j- 
d-\- .  .  .  and  into  a  second,  a-\-  bx-\-  ex*  -|-  dx*  -f- .  .  .  . 
He  perfected  what  was  already  known,  and  gave  the 
lacking  proofs  to  a  number  of  theorems,  thus  earning 
the  title  of  "founder  of  the  theory  of  combinatory 
analysis." 

The  combinatory  school,  which  included  Eschenbach,  Rothe, 
and  especially  Pfaff,  in  addition  to  its  distinguished  founder,  pro- 
duced a  varied  literature,  and  commanded  respect  because  of  its 
elegant  formal  results.  But,  in  its  aims,  it  stood  so  far  outside  the 
domain  of  the  new  and  fruitful  theories  cultivated  especially  by 
such  French  mathematicians  as  Lagrange  and  Laplace,  that  it  re- 
mained without  influence  in  the  further  development  of  mathemat- 
ics, at  least  at  the  beginning  of  the  nineteenth  century. 

In  the  domain  of  infinite  series,*  many  cases  which 
reduce  for  the  most  part  to  geometric  series,  were 
treated  by  Euclid,  and  to  a  greater  degree  by  Apol- 
lonius.  The  Middle  Ages  added  nothing  essential, 
and  it  remained  for  more  recent  generations  to  make 
important  contributions  to  this  branch  of  mathemat- 
ical knowledge.  Saint- Vincent  and  Mercator  devel- 
oped independently  the  series  for  log(l  +  *),  Gregory 
those  for  tan"1*,  sin  AT,  cos#,  sec#,  cosec^r.  In  the 
writings  of  the  latter  are  also  found,  in  the  treatment 
of  infinite  series,  the  expressions  "convergent"  and 
"divergent."  Leibnitz  was  led  to  infinite  series, 
through  consideration  of  finite  arithmetic  series.  He 
realized  at  the  same  time  the  necessity  of  examining 

*  Reiff,  R.,  Gcschichte  der  unendlichcn  Reihen,  Tubingen,  1889. 


152  HISTORY  OF  MATHEMATICS. 

more  closely  into  the  convergence  and  divergence  of 
series.  This  necessity  was  also  felt  by  Newton,  who 
used  infinite  series  in  a  manner  similar  to  that  of 
Apollonius  in  the  solution  of  algebraic  and  geometric 
problems,  especially  in  the  determination  of  areas, 
and  consequently  as  equivalent  to  integration. 

The  new  ideas  introduced  by  Leibnitz  were  further 
developed  by  Jacob  and  John  Bernoulli.  The  former 
found  the  sums  of  series  with  constant  terms,  the  lat- 
ter gave  a  general  rule  for  the  development  of  a  func- 
tion into  an  infinite  series.  At  this  time  there  were 
no  exact  criteria  for  convergence,  except  those  sug- 
gested by  Leibnitz  for  alternating  series. 

During  the  years  immediately  following,  essential 
advances  in  the  formal  treatment  of  infinite  series 
were  made.  De  Moivre  wrote  on  recurrent  series  and 
exhausted  almost  completely  their  essential  proper- 
ties. Taylor's  and  Maclaurin's  closely  related  series 
appeared,  Maclaurin  developing  a  rigorous  proof  of 
Taylor's  theorem,  giving  numerous  applications  of  it, 
and  stating  new  formulas  of  summation.  Euler  dis- 
played the  greatest  skill  in  the  handling  of  infinite 
series,  but  troubled  himself  little  about  convergence 
and  divergence.  He  deduced  the  exponential  from 
the  binomial  series,  and  was  the  first  to  develop  ra- 
tional functions  into  series  of  sines  and  cosines  of 
integral  multiple  arguments.*  In  this  manner  he 
defined  the  coefficients  of  a  trigonometric  series  by 

*  Reiff,  Geschickte  der  unenJlichen  Reihen,  1889,  pp.  105,  127. 


ALGEBRA.  1  53 

definite  integrals  without  applying  these  important 
formulas  to  the  development  of  arbitrary  functions 
into  trigonometric  series.  This  was  first  accomplished 
by  Fourier  (1822),  whose  investigations  were  com- 
pleted by  Riemann  and  Cauchy.  The  investigation 
was  brought  to  a  temporary  termination  by  Dirichlet 
(1829),  in  so  far  as  by  rigid  methods  he  gave  it  a  sci- 
entific foundation  and  introduced  general  and  com- 
plex investigations  on  the  convergence  of  series.* 
From  Laplace  date  the  developments  into  series  of 
two  variables,  especially  into  recurrent  series.  Le- 
gendre  furnished  a  valuable  extension  of  the  theory 
of  series  by  the  introduction  of  spherical  functions. 

With  Gauss  begin  more  exact  methods  of  treat- 
ment in  this  as  in  nearly  all  branches  of  mathematics, 
the  establishment  of  the  simplest  criteria  of  conver- 
gence, the  investigation  of  the  remainder,  and  the 
continuation  of  series  beyond  the  region  of  conver- 
gence. The  introduction  to  this  was  the  celebrated 
series  of  Gauss  : 


which  Euler  had  already  handled  but  whose  great 
value  he  had  not  appreciated,  f  The  generally  ac- 
cepted naming  of  this  series  as  "hypergeometric"  is 
due  to  J.  F.  Pfaff,  who  proposed  it  for  the  general 
series  in  which  the  quotient  of  any  term  divided  by  the 

*  Kuimner,  "  Gedachtnissrede  auf  Lejeune-Dirichlet."     Berliner  Abhand- 
lungen,  1860. 

t  Reiff,  Geschichtt  der  unendlicken  Reihen,  1889,  p.  161. 


154  HISTORY  OF  MATHEMATICS. 

preceding  is  a  function  of  the  index.  Euler,  follow 
ing  Wallis,  used  the  same  name  for  the  series  in  which 
that  quotient  is  an  integral  linear  function  of  the 
index.*  Gauss,  probably  influenced  by  astronomical 
applications,  stated  that  his  series,  by  assuming  cer- 
tain special  values  of  a,  /?,  y,  could  take  the  place 
of  nearly  all  the  series  then  known;  he  also  investi 
gated  the  essential  properties  of  the  function  repre- 
sented by  this  series  and  gave  for  series  in  general  an 
important  criterion  of  convergence.  We  are  indebted 
to  Abel  (1826)  for  important  investigations  on  the  con- 
tinuity of  series. 

The  idea  of  uniform  convergence  arose  from  the 
study  of  the  behavior  of  series  in  the  neighborhood  of 
their  discontinuities,  and  was  expressed  almost  simul- 
taneously by  Stokes  and  Seidel  (1847-1848).  The 
latter  calls  a  series  uniformly  convergent  when  it  rep- 
resents a  discontinuous  function  of  a  quantity  x,  the 
separate  terms  of  which  are  continuous,  but  in  the 
vicinity  of  the  discontinuities  is  of  such  a  nature  that 
values  of  x  exist  for  which  the  series  converges  as 
slowly  as  desired,  f 

On  account  of  the  lack  of  immediate  appreciation 
of  Gauss's  memoir  of  1812,  the  period  of  the  discovery 
of  effective  criteria  of  convergence  and  divergence  | 
may  be  said  to  begin  with  Cauchy  (1821).  His  meth- 

*Riemann,  Werke,  p.  78. 

tReiff,  Geschichte  der  unendlichen  Reihen,  1889,  p.  207. 
JPringsheim,  "Allgemeine  Theorie  der  Divergenz  und  Konvergenz  von 
Reihen  mil  positiven  Gliedern,"  Math.  Annalen,  XXXV. 


ALGEBRA.  155 

ods  of  investigation,  as  well  as  the  theorems  on  in- 
finite series  with  positive  terms  published  between 
1832  and  1851  by  Raabe,  Duhamel,  De  Morgan,  Ber- 
trand,  Bonnet,  and  Paucker,  set  forth  special  criteria, 
for  they  compare  generally  the  nth  term  with  particu- 
lar functions  of  the  form  a",  n*,  «(log«)*  and  others. 
Criteria  of  essentially  more  general  nature  were  first 
discovered  by  Kummer  (1835),  and  were  generalized 
by  Dini  (1867).  Dini's  researches  remained  for  a 
time,  at  least  in  Germany,  completely  unknown.  Six 
years  later  Paul  du  Bois-Reymond,  starting  with  the 
same  fundamental  ideas  as  Dini,  discovered  anew  the 
chief  results  of  the  Italian  mathematician,  worked 
them  out  more  thoroughly  and  enlarged  them  essen- 
tially to  a  system  of  convergence  and  divergence  cri- 
teria of  the  first  and  second  kind,  according  as  the 
general  term  of  the  series  an  or  the  quotient  an+I  :an  is 
the  basis  of  investigation.  Du  Bois-Reymond's  re- 
sults were  completed  and  in  part  verified  somewhat 
later  by  A.  Pringsheim. 

After  the  solution  of  the  algebraic  equations  of  the 
third  and  fourth  degrees  was  accomplished,  work  on 
the  structure  of  the  system  of  algebraic  equations  in 
general  could  be  undertaken.  Tartaglia,  Cardan,  and 
Ferrari  laid  the  keystone  of  the  bridge  which  led  from 
the  solution  of  equations  of  the  second  degree  to  the 
complete  solution  of  equations  of  the  third  and  fourth 
degrees.  But  centuries  elapsed  before  an  Abel  threw 
a  flood  of  light  upon  the  solution  of  higher  equations. 


156  HISTORY  OF  MATHEMATICS. 

Vieta  had  found  a  means  of  solving  equations  allied 
to  evolution,  and  this  was  further  developed  by  Harriot 
and  Oughtred,  but  without  making  the  process  less 
tiresome.*  Harriot's  name  is  connected  with  another 
theorem  which  contains  the  law  of  formation  of  the 
coefficients  of  an  algebraic  equation  from  its  roots, 
although  the  theorem  was  first  stated  in  full  by  Des- 
cartes (1683)  and  proved  general  by  Gauss. 

Since  there  was  lacking  a  sure  method  of  deter- 
mining the  roots  of  equations  of  higher  degree,  the 
attempt  was  made  to  include  these  roots  within  as 
narrow  limits  as  possible.  De  Beaune  and  Van 
Schooten  tried  to  do  this,  but  the  first  usable  methods 
date  from  Maclaurin  {Algebra,  published  posthum- 
ously in  1748)  and  Newton  (1722)  who  fixed  the  real 
roots  of  an  algebraic  equation  between  given  limit?. 
In  order  to  effect  the  general  solution  of  an  algebraic 
equation,  the  effort  was  made  either  to  represent  the 
given  equation  as  the  product  of  several  equations  of 
lower  degree,  a  method  further  developed  by  Hudde, 
or  to  reduce,  through  extraction  of  the  square  root, 
an  equation  of  even  degree  to  one  whose  degree  is 
half  that  of  the  given  equation  ;  this  method  was  used 
by  Newton,  but  he  accomplished  little  in  this  direc- 
tion. 

Leibnitz  had  exerted  himself  as  strenuously  as 
Newton  to  make  advances  in  the  theory  of  algebraic 
equations.  In  one  of  his  letters  he  states  that  he  has 

*  Montucla,  Histoire  des  Mathimatiques,  1799-1802. 


ALGEBRA.  157 

been  engaged  for  a  long  time  in  attempting  to  find 
the  irrational  roots  of  an  equation  of  any  degree,  by 
eliminating  the  intermediate  terms  and  reducing  it  to 
the  form  x"  =  A,  and  that  he  was,  persuaded  that  in 
this  manner  the  complete  solution  of  the  general  equa- 
tion of  the  nth  degree  could  be  effected.  This  method 
of  transformation  of  the  general  equation  dates  back 
to  Tschirnhausen  and  is  found  as  "Nova  methodus 
etc."  in  the  Leipziger  Ada  eruditorum  of  the  year  1683. 
In  the  equation 

x"  +  Ax"'1  +  BX"-*  + 
Tschirnhausen  places 


the  elimination  of  x  from  these  two  equations  gives 
likewise  an  equation  of  the  »th  degree  in  y,  in  which 
the  undetermined  coefficients  a,  ft,  y,  .  .  .  can  so  be 
taken  as  to  give  the  equation  in  y  certain  special  char- 
acteristics, for  example,  to  make  some  of  the  terms 
vanish.  From  the  values  of  y,  the  values  of  x  are  de- 
termined. By  this  method  the  solution  of  equations 
of  the  3rd  and  4th  degrees  is  made  to  depend  respec- 
tively upon  those  of  the  2nd  and  3rd  degrees  ;  but  the 
application  of  this  method  to  the  equation  of  the  5th 
degree,  leads  to  one  of  the  24th  degree,  upon  whose 
solution  the  complete  solution  of  the  equation  of  the 
5th  degree  depends. 

Afterwards,  also,  toward  the  end  of  the  seventeenth 
and  the  beginning  of  the  eighteenth  century,  De  Lagny, 


158  HISTORY  OF  MATHEMATICS. 

Rolle,  Laloubere,  and  Leseur  made  futile  attempts  to 
advance  with  rigorous  proofs  beyond  the  equation  of 
the  fourth  degree.  Euler*  took  the  problem  in  hand 
in  1749.  He  attempted  first  to  resolve  by  means  of 
undetermined  coefficients  the  equation  of  degree  2« 
into  two  equations  each  of  degree  n,  but  the  results 
obtained  by  him  were  not  more  satisfactory  than  those 
of  his  predecessors,  in  that  an  equation  of  the  eighth 
degree  by  this  treatment  led  to  an  equation  of  the  70th 
degree.  These  investigations  were  not  valueless,  how- 
ever, since  through  them  Euler  discovered  the  proof 
of  the  theorem  that  every  rational  integral  algebraic 
function  of  even  degree  can  be  resolved  into  real  fac 
tors  of  the  second  degree. 

In  a  work  of  the  date  1762  Euler  attacked  the  so- 
lution of  the  equation  of  the  nth  degree  directly.  Judg- 
ing from  equations  of  the  2nd  and  3rd  degrees,  he  sur- 
mised that  a  root  of  the  general  equation  of  the  wth 
degree  might  be  composed  of  (n —  1)  radicals  of  the 
«th  degree  with  subordinate  square  roots.  He  formed 
expressions  of  this  sort  and  sought  through  compari- 
son of  coefficients  to  accomplish  his  purpose.  This 
method  presented  no  difficulty  up  to  the  fourth  de- 
gree, but  in  the  case  of  the  fifth  degree  Euler  was 
compelled  to  limit  himself  to  particular  cases.  For 
example,  he  obtained  from 

x<>  _  40*3  _  72*8  _j_  50^  _j_  98  =  0 
the  following  value : 

*  Cantor,  III.,  p.  582. 


'  159 


7    +      _31  —  3i—  7 
_(-  l/_l8-flOi/^7  +  v/L-18  —  101/H7. 

Analogous  to  this  attempt  of  Euler  is  that  of  War- 
ing (1779).  In  order  to  solve  the  equation  /(*)  =  0 
of  degree  n,  he  places 


After  clearing  of  radicals,  he  gets  an  equation  of  the 
//th  degree,  J?(x)  =  Q,  and  by  equating  coefficients 
finds  the  necessary  equations  for  determining  a,  b,  c, 
.  .  .  q  and  p,  but  is  unable  to  complete  the  solution. 

Bdzout  also  proposed  a  method.  He  eliminated  jy 
from  the  equations  y"  —  1=0,  ay"~l  -}-  byn~*  -f-  .  .  . 
-j-^c  =  0,  and  obtained  an  equation  of  the  «th  degree, 
/(#)  =  (),  and  then  equated  coefficients.  B6zout  was 
no  more  able  to  solve  the  general  equation  of  the  5th 
degree  than  Waring,  but  the  problem  gave  him  the 
impulse  to  perfect  methods  of  elimination. 

Tschirnhausen  had  begun,  with  his  transforma- 
tion, to  study  the  roots  of  the  general  equation  as  func- 
tions of  the  coefficients.  The  same  result  can  be 
reached  by  another  method  not  different  in  principle, 
namely  the  formation  of  resolvents.  In  this  way, 
Lagrange,  Malfatti  and  Vandermonde  independently 
reached  results  which  were  published  in  the  year  1771. 
Lagrange's  work,  rich  in  matter,  gave  an  analysis  of 
all  the  then  known  methods  of  solving  equations,  and 
explained  the  difficulties  which  present  themselves  in 


l6O"  HISTORY  OF  MATHEMATICS. 

passing  beyond  the  fourth  degree.  Besides  this  he 
gave  methods  for  determining  the  limits  of  the  roots 
and  the  number  of  imaginary  roots,  as  well  as  meth- 
ods of  approximation. 

Thus  all  expedients  for  solving  the  general  equa- 
tion, made  prior  to  the  beginning  of  the  nineteenth 
century  yielded  poor  results,  and  especially  with  ref- 
erence to  Lagrange's  work  Montucla*  says  "all  this 
is  well  calculated  to  cool  the  ardor  of  those  who  are 
inclined  to  tread  this  new  way.  Must  one  entirely 
despair  of  the  solution  of  this  problem?" 

Since  the  general  problem  proved  insoluble,  at- 
tempts were  made  with  special  cases,  and  many  ele- 
gant results  were  obtained  in  this  way.  De  Moivre 
brought  the  solution  of  the  equation 


2-3.4.5   - 

for  odd  integral  values  of  n,  into  the  form 


Euler  investigated  symmetric  equations  and  Be"zout 
deduced  the  relation  between  the  coefficients  of  an 
equation  of  the  »th  degree  which  must  exist  in  order 
that  the  same  may  be  transformed  intoy-|-a  =  0. 

Gauss  made  an  especially  significant  step  in  ad- 
vance in  the  solution  of  the  cyclotomic  equation  x" —  1 
=  0,  where  n  is  a  prime  number.  Equations  of  this 
sort  are  closely  related  to  the  division  of  the  circum- 

"Hittaire  del  Sciences  Mathimatiques,  1799-1802. 


ference  into  n  equal  parts.  If  y  is  the  side  of  an  in- 
scribed «-gon  in  a  circle  of  radius  1,  and  z  the  diago- 
nal connecting  the  first  and  third  vertices,  then 


„      .         7T 

y  =  2sm—  ,    z 
n 

If  however 

27r          .    2»     /       2» 

x  =  cos  --  f-/sm  —  ,      cos 

n  n       \         n 

then  the  equation  x"  —  1  =  0  is  to  be  considered  as  the 
algebraic  expression  of  the  problem  of  the  construc- 
tion of  the  regular  «-gon. 

The  following  very  general  theorem  was  proved 
by  Gauss.*  "If  n  is  a  prime  number,  and  if  n  —  1  be 
resolved  into  prime  factors  a,  b,  c,  .  .  .  so  that  n  —  1  = 
a0-  b&  c*t  .  .  .,  then  it  is  always  possible  to  make  the  so- 
lution of  xn  —  1  =  0  depend  upon  that  of  several  equa- 
tions of  lower  degree,  namely  upon  a  equations  of 
degree  a,  /?  equations  of  degree  b,  etc."  Thus  for 
example,  the  solution  of  x7S  —  1  =  0  (the  division  of 
the  circumference  into  73  equal  parts)  can  be  effected, 
since  n  —  1=72=32.23,  by  solving  three  quadratic 
and  two  cubic  equations.  Similarly  x11  —  1=0  leads 
to  four  equations  of  the  second  degree,  since  n  —  1  = 
16  =  2*;  therefore  the  regular  17-gon  can  be  con- 
structed by  elementary  geometry,  a  fact  which  before 
the  time  of  Gauss  no  one  had  anticipated. 

Detailed  constructions  of  the  regular  17-gon  by 
elementary  geometry  were  first  given  by  Pauker  and 

*  Legendre,  Theorie  tics  Nombres. 


1 62  HISTORY  OF  MATHEMATICS. 

Erchinger.*     A  noteworthy  construction  of  the  same 
figure  is  due  to  von  Staudt. 

For  the  case  that  the  prime  number  »  has  the  form  2m  + 1, 
the  solution  of  the  equation  x"  — 1  =  0  depends  upon  the  solution 
of  m  quadratic  equations  of  which  only  m  —  1  are  necessary  in  the 
construction  of  the  regular  w-gon.  It  should  be  observed  that  for 
m  =  2*  (k  a  positive  integer),  the  number  2"1-)-!  may  be  prime, 
but,  as  R.  Baltzerf  has  pointed  out,  is  not  necessarily  prime.  If 
m  is  given  successively  the  values 

1,  2,  8.  4,  5,  6.  7,  8,  16.  212.  2", 
n  =  2**  -f- 1  will  take  the  respective  values 

3,  5,  9,  17,  33,  65,  129,  257.  65537.  2*12  +  1,  2**8 -f  1, 
of  which  only  3,  5,  17,  257,  65537  are  prime.  The  remaining  num- 
bers are  composite ;  in  particular,  the  last  two  values  of  n  have 
respectively  the  factors  114689  and  167772161.  The  circle  there- 
fore can  be  divided  into  257  or  65537  equal  parts  by  solving  re- 
spectively 7  or  15  quadratic  equations,  which  is  possible  by  ele- 
mentary geometric  construction. 

From  the  equalities 

255  =  28  — 1  =  (2*  — 1)(2*  +  1)  =   15-17,         256  =  28, 

65535  =  216  —  1  =  (28  —  1)  (28  +  1)  =  255  •  257,  65536  =  216, 
it  is  easily  seen  that,  by  elementary  geometry,  that  is,  by  use  of 
only  straight  edge  and  compasses,  the  circle  can  be  divided  respec 
tively  into  255,  256,  257  ;  65535,  65536,  65537  equal  parts.  The 
process  cannot  be  continued  without  a  break,  since  n  =232  +  1  is 
not  prime. 

The  possibility  of  an  elementary  geometric  construction  of  the 
regular  65535-gon  is  evident  from  the  following  : 
65535  =  255  •  257  =  15-17    257. 
If  the  circumference  of  the  circle  is  1,  then  since 

*  Gauss,  Werkt,  II.,  p.  187. 

t  Netto,  Substitutionentheorie,  1882 ;  English  by  Cole,  iSga,  p.  187. 


i63 


it  follows  that  gsiro  °f  tne  circumference  can  be  obtained  by  ele- 
mentary geometric  operations. 

After  Gauss  had  given  in  his  earliest  scientific 
work,  his  doctor's  dissertation,  the  first  of  his  proofs 
of  the  important  theorem  that  every  algebraic  equa- 
tion has  a  real  or  an  imaginary  root,  he  made  in  his 
great  memoir  of  1801  on  the  theory  of  numbers,  the 
conjecture  that  it  might  be  impossible  to  solve  gen- 
eral equations  of  degree  higher  than  the  fourth  by 
radicals.  Ruffini  and  Abel  gave  a  rigid  proof  of  this 
fact,  and  it  is  due  to  these  investigations  that  the 
fruitless  efforts  to  reach  the  solution  of  the  general 
equation  by  the  algebraic  method  were  brought  to  an 
end.  In  their  stead  the  question  formulated  by  Abel 
came  to  the  front,  "What  are  the  equations  of  given 
degree  which  admit  of  algebraic  solution?" 

The  cyclotomic  equations  of  Gauss  form  such  a 
group.  But  Abel  made  an  important  generalization 
by  the  theorem  that  an  irreducible  equation  is  always 
soluble  by  radicals  when  of  two  roots  one  can  be  ra- 
tionally expressed  in  terms  of  the  other,  provided  at 
the  same  time  the  degree  of  the  equation  is  prime ;  if 
this  is  not  the  case,  the  solution  depends  upon  the 
solution  of  equations  of  lower  degree. 

A  further  great  group  of  algebraically  soluble  equa- 
tions is  therefore  comprised  in  the  Abelian  equations. 
But  the  question  as  to  the  necessary  and  sufficient 
conditions  for  the  algebraic  solubility  of  an  equation 


164  HISTORY  OF  MATHEMATICS. 

was  first  answered  by  the  youthful  Galois,  the  crown 
of  whose  investigations  is  the  theorem,  "If  the  degree 
of  an  irreducible  equation  is  a  prime  number,  the 
equation  is  soluble  by  radicals  alone,  provided  the 
roots  of  this  equation  can  be  expressed  rationally  in 
terms  of  any  two  of  them." 

Abel's  investigations  fall  between  the  years  1824 
and  1829,  those  of  Galois  in  the  years  1830  and  1831. 
Their  fundamental  significance  for  all  further  labors 
in  this  direction  is  an  undisputed  fact ;  the  question 
concerning  the  general  type  of  algebraically  soluble 
equations  alone  awaits  an  answer. 

Galois,  who  also  earned  special  honors  in  the  field 
of  modular  equations  which  enter  into  the  theory  of 
elliptic  functions,  introduced  the  idea  of  a  group  of 
substitutions.*  The  importance  of  this  innovation, 
and  its  development  into  a  formal  theory  of  substitu- 
tions, as  Cauchy  has  first  given  it  in  the  Exercices 
d }  analyse,  etc.,  when  he  speaks  of  "systems  of  con- 
jugate substitutions,"  became  manifest  through  geo- 
metric considerations.  The  first  example  of  this  was 
furnished  by  Hesse f  in  his  investigation  on  the  nine 
points  of  inflexion  of  a  curve  of  the  third  degree.  The 
equation  of  the  ninth  degree  upon  which  they  depend 
belongs  to  the  class  of  algebraically  soluble  equations. 
In  this  equation  there  exists  between  any  two  of  the 
roots  and  a  third  determined  by  them  an  algebraic  re- 

*Netto,  Subititutionentheorie,  1882.     English  by  Cole,  1892. 
+  Noether,  O.  H.,  Schlomilch's  Zeitschrift,  Band  20. 


ALGEBRA.  165 

lation  expressing  the  geometric  fact  that  the  nine 
points  of  inflexion  lie  by  threes  on  twelve  straight 
lines.  To  the  development  of  the  substitution  theory 
in  later  times,  Kronecker,  Klein,  Noether,  Hermite, 
Betti,  Serret,  Poincar6,  Jordan,  Capelli,  and  Sylow 
especially  have  contributed. 

Most  of  the  algebraists  of  recent  times  have  par- 
ticipated in  the  attempt  to  solve  the  equation  of  the 
fifth  degree.  Before  the  impossibility  of  the  algebraic 
solution  was  known,  Jacobi  at  the  age  of  16  had  made 
an  attempt  in  this  direction ;  but  an  essential  advance 
is  to  be  noted  from  the  time  when  the  solution  of  the 
equation  of  the  fifth  degree  was  linked  with  the  theory 
of  elliptic  functions.*  By  the  help  of  transformations 
as  given  on  the  one  hand  by  Tschirnhausen  and  on 
the  other  by  E.  S.  Bring  (1786),  the  roots  of  the  equa- 
tion of  the  fifth  degree  can  be  made  to  depend  upon 
a  single  quantity  only,  and  therefore  the  equation,  as 
shown  by  Hermite,  can  be  put  into  the  form  t* —  / — A 
=  0.  By  Riemann's  methods,  the  dependence  of  the 
roots  of  the  equation  upon  the  parameter  A  is  illus- 
trated; on  the  other  hand,  it  is  possible  by  power- 
series  to  calculate  these  five  roots  to  any  degree  of  ap- 
proximation. In  1858,  Hermite  and  Kronecker  solved 
the  equation  of  the  fifth  degree  by  elliptic  functions, 
but  without  reference  to  the  algebraic  theory  of  this 
equation,  while  Klein  gave  the  simplest  possible  solu- 

*  Klein,    F.,    Vergleichende   Betrachtvitgcn    uber   neuere  geomeirische  For- 
scliunScn,  1872. 


l66  HISTORY  OF  MATHEMATICS. 

tion  by  transcendental  functions  by  using  the  theory 
of  the  icosahedron. 

The  solution  of  general  equations  of  the  nth  degree  for  »>4 
by  transcendental  functions  has  therefore  become  possible,  and 
the  operations  entering  into  the  solution  are  the  following :  Solu- 
tion of  equations  of  lower  degree ;  solution  of  linear  differential 
equations  with  known  singular  points ;  determination  of  constants 
of  integration,  by  calculating  the  moduli  of  periodicity  of  hyper- 
elliptic  integrals  for  which  the  branch-points  of  the  function  to  be 
integrated  are  known  ;  finally  the  calculation  of  theta-functions  of 
several  variables  for  special  values  of  the  argument. 

The  methods  leading  to  the  complete  solution  of 
an  algebraic  equation  are  in  many  cases  tedious ;  on 
this  account  the  methods  of  approximation  of  real 
roots  are  very  important,  especially  where  they  can 
be  applied  to  transcendental  equations.  The  most 
general  method  of  approximation  is  due  to  Newton 
(communicated  to  Barrow  in  1669),  but  was  also 
reached  by  Halley  and  Raphson  in  another  way.* 
For  the  solution  of  equations  of  the  third  and  fourth 
degrees,  John  Bernoulli  worked  out  a  valuable  method 
of  approximation  in  his  Lectiones  calculi  integralis. 
Further  methods  of  approximation  are  due  to  Daniel 
Bernoulli,  Taylor,  Thomas  Simpson,  Lagrange,  Le- 
gendre,  Homer,  and  others. 

By  graphic  and  mechanical  means  also,  the  roots  of  an  equa- 
tion can  be  approximated.     C.  V.  Boysf  made  use  of  a  machine 
for  this  purpose,  which  consisted  of  a  system  of  levers  and  ful- 
crums  ;  Cunynghamef  used  a  cubic  parabola  with  a  tangent  scale 
*Montucla.  t  Nature,  XXXIII.,  p.  166 


1 67 


on  a  straight  edge ;  C.  Reuschle*  used  an  hyperbola  with  an  ac- 
companying gelatine-sheet,  so  that  the  roots  could  be  read  as  in- 
tersections of  an  hyperbola  with  a  parabola.  Similar  methods, 
suited  especially  to  equations  of  the  third  and  fourth  degrees  are 
due  to  Bartl,  R.  Hoppe,  and  Oekinghausf ;  Lalanne  and  Mehmke 
also  deserve  mention  in  this  connection. 

For  the  solution  of  equations,  there  had  been  in- 
vented in  the  seventeenth  century  an  algorism  which 
since  then  has  gained  a  place  in  all  branches  of  mathe- 
matics, the  algorism  of  determinants.  |  The  first  sug- 
gestion of  computation  with  those  regularly  formed 
aggregates,  which  are  now  called  determinants  (after 
Cauchy),  was  given  by  Leibnitz  in  the  year  1693. 
He  used  the  aggregate 

an,  a\i, a\* 

<*2i,  #22, a** 


in  forming  the  resultant  of  «  linear  equations  with 
« — 1  unknowns,  and  that  of  two  algebraic  equations 
with  one  unknown.  Cramer  (1750)  is  considered  as 
a  second  inventor,  because  he  began  to  develop  a  sys- 
tem of  computation  with  determinants.  Further  the- 
orems are  due  to  Bezout  (1764),  Vandermonde  (1771), 
Laplace  (1772),  and  Lagrange  (1773).  Gauss's  Dis- 
quisitiones  arithmeticae  (1801)  formed  an  essential  ad- 

*  BBklen,  O.,  Math.  Mittheilwngen,  1886,  p.  IO3. 
t  Fortschritte,  1883;  1884. 

t  Muir,  T.,  Theory  of  Determinant*  in  the  Historical  Order  of  its  Develop- 
ment, Parti,  1890;  Baltzer.  R.,  Theorie  und  Antvendungen  der  Determinanten, 


l68  HISTORY   OF  MATHEMATICS. 

vance,  and  this  gave  Cauchy  the  impulse  to  many 
new  investigations,  especially  the  development  of  the 
general  law  (1812)  of  the  multiplication  of  two  deter- 
minants. 

Jacobi  by  his  "masterful  skill  in  technique,"  also 
rendered  conspicuous  service  in  the  theory  of  determi- 
nants, having  developed  a  theory  of  expressions  which 
he  designated  as  "functional  determinants."  The 
analogy  of  these  determinants  with  differential  quo- 
tients led  him  to  the  general  "principle  of  the  last 
multiplier  "  which  plays  a  part  in  nearly  all  problems 
of  integration.*  Hesse  considered  in  an  especially 
thorough  manner  symmetric  determinants  whose  ele- 
ments are  linear  functions  of  the  co-ordinates  of  a 
geometric  figure.  He  observed  their  behavior  by  lin- 
ear transformation  of  the  variables,  and  their  rela- 
tions to  such  determinants  as  are  formed  from  them 
by  a  single  bordering."!'  Later  discussions  are  due  to 
Cayley  on  skew  determinants,  and  to  Nachreiner  and 
S.  Giinther  on  relations  between  determinants  and 
continued  fractions. 

The  appearance  of  the  differential  calculus  forms 
one  of  the  most  magnificent  discoveries  of  this  period. 
The  preparatory  ideas  for  this  discovery  appear  in 
manifest  outline  in  Cavalieri,J  who  in  a  work  Metho- 
dus  indivisibilium  (1635)  considers  a  space- element  as 

*  Dirichlet,  "  Gedachtnissrede  auf  Jacobi."     Crelle's  Journal,  Band  52. 
tNoether,  O.  H.,  Schlomilch's  Zeittchrift,  Band  20. 
tl.iiroth,  Rektoratsrede,  Freiburg,  1889;  Cantor,  II.,  p.  759. 


ALGEBRA.  169 

the  sum  of  an  infinite  number  of  simplest  space-ele- 
ments of  the  next  lower  dimension,  e.  g.,  a^solid  as 
the  sum  of  an  infinite  number  of  planes.  The  danger 
of  this  conception  was  fully  appreciated  by  the  inven- 
tor of  the  method,  but  it  was  improved  first  by  Pascal 
who  considers  a  surface  as  composed  of  an  infinite 
number  of  infinitely  small  rectangles,  then  by  Fermat 
and  Roberval ;  in  all  these  methods,  however,  there 
appeared  the  drawback  that  the  sum  of  the  resulting 
series  could  seldom  be  determined.  Kepler  remarked 
that  a  function  can  vary  only  slightly  in  the  vicinity 
of  a  greatest  or  least  value.  Fermat,  led  by  this 
thought,  made  an  attempt  to  determine  the  maximum 
or  minimum  of  a  function.  Roberval  investigated  the 
problem  of  drawing  a  tangent  to  a  curve,  and  solved 
it  by  generating  the  curved  line  by  the  composition  of 
two  motions,  and  applied  the  parallelogram  of  veloci- 
ties to  the  construction  of  the  tangents.  Barrow, 
Newton's  teacher,  used  this  preparatory  work  with 
reference  to  Cartesian  co-ordinate  geometry.  He 
chose  the  rectangle  as  the  velocity- parallelogram,  and 
at  the  same  time  introduced  like  Fermat  infinitely 
small  quantities  as  increments  of  the  dependent  and 
independent  variables,  with  special  symbols.  He  gave 
also  the  rule,  that,  without  affecting  the  validity  of  the 
result  of  computation,  higher  powers  of  infinitely  small 
quantities  may  be  neglected  in  comparison  with  the 
first  power.  But  Barrow  was  not  able  to  handle  frac- 
tions and  radicals  involving  infinitely  small  quantities, 


170  HISTORY  OF  MATHEMATICS. 

and  was  compelled  to  resort  to  transformations  to  re- 
move them.  Like  his  predecessors,  he  was  able  to 
determine  in  the  simpler  cases  the  value  of  the  quo- 
tient of  two,  or  the  sum  of  an  infinite  number  of  in- 
finitesimals. The  general  solution  of  such  questions 
was  reached  by  Leibnitz  and  Newton,  the  founders  of 
the  differential  calculus. 

Leibnitz  gave  for  the  calculus  of  infinitesimals,  the 
notion  of  which  had  been  already  introduced,  further 
examples  and  also  rules  for  more  complicated  cases. 
By  summation  according  to  the  old  methods,*  he  de- 
duced the  simplest  theorems  of  the  integral  calculus, 
which  he,  by  prefixing  a  long  S  as  the  sign  of  summa- 
tion wrote, 

/— £  /*=£  /<•+»=/-+/'• 

From  the  fact  that  the  sign  of  summation  C  raised 
the  dimension,  he  drew  the  conclusion  that  by  differ- 
ence-forming the  dimension  must  be  diminished  so 
that,  therefore,  as  he  wrote  in  a  manuscript  of  Oct. 

29,  1675,  from    Cl=ya,  follows  immediately  1=-^. 
O  a 

Leibnitz  tested  the  power  of  his  new  method  by 
geometric  problems ;  he  sought,  for  example,  to  de- 
termine the  curve  "for  which  the  intercepts  on  the 
axis  to  the  feet  of  the  normals  vary  as  the  ordinates." 
In  this  he  let  the  abscissas  x  increase  in  arithmetic 
ratio  and  designated  the  constant  difference  of  the 

•Gerhardt,  Geschichte  der  Mathematik  in  Devttckland,  1877;  Cantor,  III., 
p.  160. 


abscissas  first  by  —  and  later  by  dx,  without  explain- 
ing in  detail  the  meaning  of  this  new  symbol.  In 
1676  Leibnitz  had  developed  his  new  calculus  so  far 
as  to  be  able  to  solve  geometric  problems  which  could 
not  be  reduced  by  other  methods.  Not  before  1686, 
however,  did  he  publish  anything  about  his  method, 
its  great  importance  being  then  immediately  recog- 
nized by  Jacob  Bernoulli. 

What  Leibnitz  failed  to  explain  in  the  develop- 
ment of  his  methods,  namely  what  is  understood  by 
his  infinitely  small  quantities,  was  clearly  expressed 
by  Newton,  and  secured  for  him  a  theoretical  superi- 
ority. Of  a  quotient  of  two  infinitely  small  quantities 
Newton  speaks  as  of  a  limiting  value*  which  the  ratio 
of  the  vanishing  quantities  approaches,  the  smaller 
they  become.  Similar  considerations  hold  for  the  sum 
of  an  infinite  number  of  such  quantities.  For  the  de- 
termination of  limiting  values,  Newton  devised  an 
especial  algorism,  the  calculus  of  fluxions,  which  is 
essentially  identical  with  Leibnitz's  differential  calcu- 
lus. Newton  considered  the  change  in  the  variable 
as  a  flowing ;  he  sought  to  determine  the  velocity  of 
the  variation  of  the  function  when  the  variable  changes 
with  a  given  velocity.  The  velocities  were  called 
fluxions  and  were  designated  by  x,  y,  z  (instead  of 
dx,  dy,  dz,  as  in  Leibnitz's  writings).  The  quantities 
themselves  were  called  fluents,  and  the  calculus  of 
fluxions  determines  therefore  the  velocities  of  given 

*  Liiroth,  Rektoratsrede,  Freiburg,  1889. 


172  HISTORY  OF  MATHEMATICS. 

motions,  or  seeks  conversely  to  find  the  motions  when 
the  law  of  their  velocities  is  known.  Newton's  paper 
on  this  subject  was  finished  in  1671  under  the  name 
of  Methodus  fluxionum,  but  was  first  published  in  1736, 
after  his  death.  Newton  is  thought  by  some  to  have 
borrowed  the  idea  of  fluxions  from  a  work  of  Napier.* 
According  to  Gauss,  Newton  deserved  much  more 
credit  than  Leibnitz,  although  he  attributes  to  the 
latter  great  talent,  which,  however,  was  too  much  dis- 
sipated. It  appears  that  this  judgment,  looked  at 
from  both  sides,  is  hardly  warranted.  Leibnitz  failed 
to  give  satisfactory  explanation  of  that  which  led 
Newton  to  one  of  his  most  important  innovations,  the 
idea  of  limits.  On  the  other  hand,  Newton  is  not 
always  entirely  clear  in  the  purely  analytic  proo  . 
Leibnitz,  too,  deserves  very  high  praise  for  the  intro- 
duction of  the  appropriate  symbols  C  and  dx,  as  well 
as  for  stating  the  rules  of  operating  with  them.  To- 
day the  opinion  might  safely  be  expressed  that  the 
differential  and  integral  calculus  was  independently 
discovered  by  Newton  and  by  Leibnitz ;  that  Newton 
is  without  doubt  the  first  inventor;  that  Leibnitz,  on 
the  other  hand,  stimulated  by  the  results  communi- 
cated to  him  by  Newton,  but  without  the  knowledge 
of  Newton's  methods,  invented  independently  the 
calculus;  and  that  finally  to  Leibnitz  belongs  the 
priority  of  publication,  "f 

*  Cohen,  Dot  Frinnip  der  lufi nitesimalmethodc  und  seine  Getckichte,  1889: 
Cantor,  III.,  p.  163. 

+  Lfiroth.    A  very  good  summary  of  the  discussion  is  also  given  in  Balls 


ALGEBRA.  173 

The  systematic  development  of  the  new  calculus 
made  necessary  a  clearer  understanding  of  the  idea  of 
the  infinite.  Investigations  on  the  infinitely  great  are 
of  course  of  only  passing  interest  for  the  explanation 
of  natural  phenomena,*  but  it  is  entirely  different 
with  the  question  of  the  infinitely  small.  The  infini- 
tesimal f  appears  in  the  writings  of  Kepler  as  well  as 
in  those  of  Cavalieri  and  Wallis  under  varying  forms, 
essentially  as  "infinitely  small  null- value,"  that  is,  as 
a  quantity  which  is  smaller  than  any  given  quantity, 
and  which  forms  the  limit  of  a  given  finite  quantity. 
Euler's  indivisibilia  lead  systematically  in  the  same 
direction.  Fermat,  Roberval,  Pascal,  and  especially 
Leibnitz  and  Newton  operated  with  the  "unlimitedly 
small,"  yet  in  such  a  way  that  frequently  an  abbrevi- 
ated method  of  expression  concealed  or  at  least  ob- 
scured the  true  sense  of  the  development.  In  the 
writings  of  John  Bernoulli,  De  1'Hospital,  and  Pois- 
son,  the  infinitesimal  appears  as  a  quantity  different 
from  zero,  but  which  must  become  less  than  an  assign- 
able value,  i.  e.,  as  a  "  pseudo-infinitesimal  "  quantity. 
By  the  formation  of  derivatives,  which  in  the  main 
are  identical  with  Newton's  fluxions,  LagrangeJ  at- 
tempted entirely  to  avoid  the  infinitesimal,  but  his 
attempts  only  served  the  purpose  of  bringing  into 

Short  History  of  Mathematics,  London,  1888.    The  best  summary  is  that  given 
in  Cantor,  Vol.  III. 

*  Riemann,  Werke,  p.  267. 

t  R.  Hoppe,  Differentialrechnung,  1865. 

tLiiroth,  Rekioratsrede,  Freiburg,  1889. 


174  HISTORY  OF  MATHEMATICS. 

prominence  the  urgent  need  for  a  deeper  foundation 
for  the  theory  of  the  infinitesimal  for  which  Tacquet 
and  Pascal  in  the  seventeenth  century,  and  Maclaurin 
and  Carnot  in  the  eighteenth  had  made  preparation. 
We  are  indebted  to  Cauchy  for  this  contribution.  In 
his  investigations  there  is  clearly  established  the  mean- 
ing of  propositions  which  contain  the  expression  "in- 
finitesimal," and  a  safe  foundation  for  the  differential 
calculus  is  thereby  laid. 

The  integral  calculus  was  first  further  extended 
by  Cotes,  who  showed  how  to  integrate  rational  alge- 
braic functions.  Legendre  applied  himself  to  the  in- 
tegration of  series,  Gauss  to  the  approximate  deter- 
mination of  integrals,  and  Jacobi  to  the  reduction  and 
evaluation  of  multiple  integrals.  Dirichlet  is  espe- 
cially to  be  credited  with  generalizations  on  definite 
integrals,  his  lectures  showing  his  great  fondness  for 
this  theory.*  He  it  was  who  welded  the  scattered 
results  of  his  predecessors  into  a  connected  whole, 
and  enriched  them  by  a  new  and  original  method  of 
integration.  The  introduction  of  a  discontinuous  fac- 
tor allowed  him  to  replace  the  given  limits  of  integra- 
tion by  different  ones,  often  by  infinite  limits,  without 
changing  the  value  of  the  integral.  In  the  more  re 
cent  investigations  the  integral  has  become  the  means 
of  defining  functions  or  of  generating  others. 

In  the  realm  of  differential  equations  f  the  works 

*Knmmer,  " Gedachtnissrede  auf  Lejeune-Dirichlet."  Berliner  Abh.,  1860 
t  Cantor,  III.,  p.  429;  Schiesinger,  L.,  Handbuch  dtr  Tkeorie  der  linearen 


ALGEBRA.  175 

worthy  of  mention  date  back  to  Jacob  and  John  Ber- 
noulli and  to  Riccati.  Riccati's  merit  consists  mainly 
in  having  introduced  Newton's  philosophy  into  Italy. 
He  also  integrated  for  special  cases  the  differential 
equation  named  in  his  honor — an  equation  completely 
solved  by  Daniel  Bernoulli — and  discussed  the  ques- 
tion of  the  possibility  of  lowering  the  order  of  a  given 
differential  equation.  The  theory  first  received  a  de- 
tailed scientific  treatment  at  the  hands  of  Lagrange, 
especially  as  far  as  concerns  partial  differential  equa- 
tions, of  which  D'Alembert  and  Kuler  had  handled 

d"*u        d^u 
the  equation  —^  =  —r^-     Laplace  also  wrote  on  this 

differential  equation  and  on  the  reduction  of  the  solu- 
tion of  linear  differential  equations  to  definite  integ- 
rals. 

On  German  soil,  J.  F.  Pfaff,  the  friend  of  Gauss 
and  next  to  him  the  most  eminent  mathematician 
of  that  time,  presented  certain  elegant  investigations 
(1814,  1815)  on  differential  equations,*  which  led 
Jacobi  to  introduce  the  name  "Pfaffian  problem." 
Pfaff  found  in  an  original  way  the  general  integration 
of  partial  differential  equations  of  the  first  degree  for 
any  number  of  variable  quantities.  Beginning  with 
the  theory  of  ordinary  differential  equations  of  the 
first  degree  with  n  variables,  for  which  integrations 

Differentialgleichnngen,  Bd.  I.,  1895, — an  excellent  historical  review;  Mansion, 
P.,  Theorie  der  partiellen  Dlfferentialgleichungen  erster  Ordnung,  deutsch 
von  Maser,  Leipzig,  1892,  also  excellent  on  history. 

*A.  Brill,  "Das  mathematisch-physikalische  Seminar  in  Tubingen." 
Aus  der  Festschrift  der  Uni-versitat  turn  KSnigs-JMliium,  1889. 


176  HISTORY  OF  MATHEMATICS. 

were  given  by  Monge  (1809)  in  special  simple  cases, 
Pfaff  gave  their  general  integration  and  considered 
the  integration  of  partial  differential  equations  as  a 
particular  case  of  the  general  integration.  In  this  the 
general  integration  of  differential  equations  of  every 
degree  between  two  variables  is  assumed  as  known.* 
Jacobi  (1827, 1836)  also  advanced  the  theory  of  differ- 
ential equations  of  the  first  order.  The  treatment 
was  so  to  determine  unknown  functions  that  an  integ- 
ral which  contains  these  functions  and  the  differential 
coefficient  in  a  prescribed  way  reaches  a  maximum  or 
minimum.  The  condition  therefor  is  the  vanishing  of. 
the  first  variation  of  the  integral,  which  again  finds  its 
expression  in  differential  equations,  from  which  the 
unknown  functions  are  determined.  In  order  to  be 
able  to  distinguish  whether  a  real  maximum  or  mini- 
mum appears,  it  is  necessary  to  bring  the  second  va- 
riation into  a  form  suitable  for  investigating  its  sign. 
This  leads  to  new  differential  equations  which  La- 
grange  was  not  able  to  solve,  but  of  which  Jacobi  was 
able  to  show  that  their  integration  can  be  deduced 
from  the  integration  of  differential  equations  belong- 
ing to  the  first  variation.  Jacobi  also  investigated 
the  special  case  of  a  simple  integral  with  one  unknown 
function,  his  statements  being  completely  proved  by 
Hesse.  Clebsch  undertook  the  general  investigation 
of  the  second  variation,  and  he  was  successful  in 
showing  for  the  case  of  multiple  integrals  that  new 

*  Gauss,  Wtrke,  III.,  p.  232. 


ALGEBRA.  177 

integrals  are  not  necessary  for  the  reduction  of  the 
second  variation.  Clebsch  (1861,  1862),  following  the 
suggestions  of  Jacobi,  also  reached  the  solution  of  the 
Pfaffian  problem  by  making  it  depend  upon  a  system 
of  simultaneous  linear  partial  differential  equations, 
the  statement  of  which  is  possible  without  integration. 
Of  other  investigations,  one  of  the  most  important  is 
the  theory  of  the  equation 

^!l'  +  ^  +  ^=o, 

which  Dirichlet  encountered  in  his  work  on  the  po- 
tential, but  which  had  been  known  since  Laplace 
(1789).  Recent  investigations  on  differential  equa- 
tions, especially  on  the  linear  by  Fuchs,  Klein,  and 
Poincare,  stand  in  close  connection  with  the  theories 
of  functions  and  groups,  as  well  as  with  those  of  equa- 
tions and  series. 

"Within  a  half  century  the  theory  of  ordinary  differential 
equations  has  come  to  be  one  of  the  most  important  branches  of 
analysis,  the  theory  of  partial  differential  equations  remaining  as 
one  still  to  be  perfected.  The  difficulties  of  the  general  problem 
of  integration  are  so  manifest  that  all  classes  of  investigators  have 
confined  themselves  to  the  properties  of  the  integrals  in  the  neigh- 
borhood of  certain  given  points.  The  new  departure  took  its 
greatest  inspiration  from  two  memoirs  by  Fuchs  (1866,  1868),  a 
work  elaborated  by  Thome"  and  Frobenius.  .  .  . 

"Since  1870  Lie's  labors  have  put  the  entire  theory  of  differ- 
ential equations  on  a  more  satisfactory  foundation.  He  has  shown 
that  the  integration  theories  of  the  older  mathematicians,  which 
had  been  looked  upon  as  isolated,  can  by  the  introduction  of  the 
concept  of  continuous  groups  of  transformations  be  referred  to  a 


178  HISTORY  OF  MATHEMATICS. 

common  source,  and  that  ordinary  differential  equations  which 
admit  the  same  infinitesimal  transformations  present  like  difficul- 
ties of  integration  He  has  also  emphasized  the  subject  of  trans- 
formations of  contact  (Beriihrungs-Transformationen)  which 
underlies  so  much  of  the  recent  theory.  .  .  .  Recent  writers  have 
shown  the  same  tendency  noticeable  in  the  works  of  Monge  and 
Cauchy,  the  tendency  to  separate  into  two  schools,  the  one  inclin- 
ing to  use  the  geometric  diagram  and  represented  by  Schwarz, 
Klein,  and  Goursat,  the  other  adhering  to  pure  analysis,  of  which 
Weierstrass,  Fuchs,  and  Frobenius  are  types."* 

A  short  time  after  the  discovery  of  the  differential 
and  integral  calculus,  namely  in  the  year  1696,  John 
Bernoulli  proposed  this  problem  to  the  mathemati- 
cians of  his  time :  To  find  the  curve  described  by  a 
body  falling  from  a  given  point  A  to  another  given 
point  B  in  the  shortest  time.f  The  problem  came  from 
a  case  in  optics,  and  requires  a  function  to  be  found 
whose  integral  is  a  minimum.  Huygens  had  devel- 
oped the  wave-theory  of  light,  and  John  Bernoulli 
had  found  under  definite  assumptions  the  differential 
equation  of  the  path  of  the  ray  of  light.  Of  such  mo- 
tion he  sought  another  example,  and  came  upon  the 
cycloid  as  the  brachistochrone,  that  is,  upon  the  above 
statement  of  the  problem,  for  which  up  to  Easter 
1697,  solutions  from  the  Marquis  de  1'Hospital,  from 
Tschirnhausen,  Newton,  Jacob  Bernoulli  and  Leib- 
nitz were  received.  Only  the  two  latter  treated  the 

*  Smith,  D.  E.,  "History  of  Modern  Mathematics,"  in  Merriman  and 
Woodward's  Higher  Mathematics,  New  York,  1896,  with  authorities  cited. 

t  Reiff ,  R.,  "  Die  Anfange  der  Variationsrechnung,"  Math.  Mittheilungen 
von  Boklen,  1887.  Cantor,  III.,  p.  225.  Woodhonse,  A  Treatise  on  Isoferimet- 
rical  Problems  (Cambridge,  1810).  The  last  named  work  is  rare. 


ALGEBRA.  179 

problem  as  one  of  maxima  and  minima.  Jacob  Ber- 
noulli's method  remained  the  common  one  for  the 
treatment  of  similar  cases  up  to  the  time  of  Lagrange, 
and  he  is  therefore  to  be  regarded  as  one  of  the  found- 
ers of  the  calculus  of  variations.  At  that  time*  all 
problems  which  demanded  the  statement  of  a  maxi- 
mum or  minimum  property  of  functions  were  called 
isoperimetric  problems.  To  the  oldest  problems  of 
this  kind  belong  especially  those  in  which  one  curve 
with  a  maximum  or  minimum  property  was  to  be  found 
from  a  class  of  curves  of  equal  perimeters.  That  the 
circle,  of  all  isoperimetric  figures,  gives  the  maximum 
area,  is  said  to  have  been  known  to  Pythagoras.  In 
the  writings  of  Pappus  a  series  of  propositions  on  fig- 
ures of  equal  perimeters  are  found.  Also  in  the  four- 
teenth century  the  Italian  mathematicians  had  worked 
on  problems  of  this  kind.  But  "the  calculus  of  varia- 
tions may  be  said  to  begin  with  .  .  .  John  Bernoulli 
(1696).  It  immediately  occupied  the  attention  of 
Jacob  Bernoulli  and  the  Marquis  de  PHospital,  but 
Euler  first  elaborated  the  subject,  "f  He|  investigated 
the  isoperimetric  problem  first  in  the  analytic-geo- 
metric manner  of  Jacob  Bernoulli,  but  after  he  had 
worked  on  the  subject  eight  years,  he  came  in  1744 
upon  a  new  and  general  solution  by  a  purely  analytic 
method  (in  his  celebrated  work  :  Methodus  inveniendi 

*  Anton,  Geschichte  des  isoperimetrischen  Problems,  1888. 
t  Smith,  D.  E.,  History  of  Modern  Mathematics,  p.  533. 
$  Cantor,  III.,  pp.  243,  819,  830. 


l8o  HISTORY  OF  MATHEMATICS. 

lineas  curvas,  etc.);  this  solution  shows  how  those  or- 
dinates  of  the  function  which  are  to  assume  a  greatest 
or  least  value  can  be  derived  from  the  variation  of  the 
curve-ordinate.  Lagrange  (£ssai  d'une  nouvelle  m^ 
thode,  etc.,  1760  and  1761)  made  the  last  essential  step 
from  the  pointwise  variation  of  Euler  and  his  prede- 
cessors to  the  simultaneous  variation  of  all  ordinates 
of  the  required  curve  by  the  assumption  of  variable 
limits  of  the  integral.  His  methods,  which  contained 
the  new  feature  of  introducing  8  for  the  change  of  the 
function,  were  later  taken  up  in  Eider's  Integral  Cal- 
culus. Since  then  the  calculus  of  variations  has  been 
of  valuable  service  in  the  solution  of  problems  in  the- 
ory of  curvature. 

The  beginnings  of  a  real  theory  of  functions*,  espe- 
cially that  of  the  elliptic  and  Abelian  functions  lead 
back  to  Fagnano,  Maclaurin,  D'Alembert,  and  Landen. 
Integrals  of  irrational  algebraic  functions  were  treated, 
especially  those  involving  square  roots  of  polynomials 
of  the  third  and  fourth  degrees ;  but  none  of  these 
works  hinted  at  containing  the  beginnings  of  a  science 
dominating  the  whole  subject  of  algebra.  The  matter 
assumed  more  definite  form  under  the  hands  of  Euler, 
Lagrange,  and  Legendre.  For  a  long  time  the  only 
transcendental  functions  known  were  the  circular  func- 


*  Brill,  A.,  and  Noether,  M.,  "Die  Entwickelnng  der  Theorie  der  alge- 
braischen  Functionen  in  atterer  und  neuerer  Zeit,  Bericht  erstattet  der  Deut- 
schen  Mathematiker-Vereinigung,  Jakresbertcht,  Bd.  II.,  pp.  107-566,  Berlin, 
1894  ;  KOnigsberger,  L.,  Zur  Geschichte  der  Theorie  der  elliptischen  Transcen- 
denten  in  den  Jahren  1826-1829,  Leipzig,  1879. 


ALGEBRA.  l8l 

tions  (sin  x,  cos  x,  .  .  .),  the  common  logarithm,  and, 
especially  for  analytic  purposes,  the  hyperbolic  log- 
arithm with  base  e,  and  (contained  in  this)  the  ex- 
ponential function  e*.  But  with  the  opening  of  the 
nineteenth  century  mathematicians  began  on  the  one 
hand  thoroughly  to  study  special  transcendental  func- 
tions, as  was  done  by  Legendre,  Jacobi,  and  Abel, 
and  on  the  other  hand  to  develop  the  general  theory 
of  functions  of  a  complex  variable,  in  which  field 
Gauss,  Cauchy,  Dirichlet,  Riemann,  Liouville,  Fuchs, 
and  Weierstrass  obtained  valuable  results. 

The  first  signs  of  an  interest  in  elliptic  functions* 
are  connected  with  the  determination  of  the  arc  of  the 
lemniscate,  as  this  was  carried  out  in  the  middle  of 
the  eighteenth  century.  In  this  Fagnano  made  the 
discovery  that  between  the  limits  of  two  integrals  ex- 
pressing the  arc  of  the  curve,  one  of  which  has  twice 
the  value  of  the  other,  there  exists  an  algebraic  rela- 
tion of  simple  nature.  By  this  means,  the  arc  of  the 
lemniscate,  though  a  transcendent  of  higher  order, 
can  be  doubled  or  bisected  by  geometric  construc- 
tion like  an  arc  of  a  circle,  f  Euler  gave  the  ex- 
planation of  this  remarkable  phenomenon.  He  pro- 
duced a  more  general  integral  than  Fagnano  (the 
so-called  elliptic  integral  of  the  first  class)  and  showed 
that  two  such  integrals  can  be  combined  into  a  third 
of  the  same  kind,  so  that  between  the  limits  of  these 

*Enneper,  A.,  Elliptische  Function™,  Theorie  und  Geschichtc,  Halle,  1890. 
t  Dirichlet,  "  Gedachtnissrede  auf  Jacobi."     Crelle's  Journal,  Bd.  52. 


l82  HISTORY  OF  MATHEMATICS. 

integrals  there  exists  a  simple  algebraic  relation,  just 
as  the  sine  of  the  sum  of  two  arcs  can  be  composed  of 
the  same  functions  of  the  separate  arcs  (addition-the- 
orem). The  elliptic  integral,  however,  depends  not 
merely  upon  the  limits  but  upon  another  quantity  be- 
longing to  the  function,  the  modulus.  While  Euler 
placed  only  integrals  with  the  same  modulus  in  rela- 
tion, Landen  and  Lagrange  considered  those  with 
different  moduli,  and  showed  that  it  is  possible  by 
simple  algebraic  substitution  to  change  one  elliptic 
integral  into  another  of  the  same  class.  The  estab- 
lishment of  the  addition-theorem  will  always  remain 
at  least  as  important  a  service  of  Euler  as  his  trans- 
formation of  the  theory  of  circular  functions  by  the 
introduction  of  imaginary  exponential  quantities. 

The  origin*  of  the  real  theory  of  elliptic  functions 
and  the  theta-functions  falls  between  1811  and  1829. 
To  Legendre  are  due  two  systematic  works,  the  Exer- 
cices  de  calcul  integral  (1811-1816)  and  the  Thtorie  des 
functions  elliptiques  (1825-1828),  neither  of  which  was 
known  to  Jacobi  and  Abel.  Jacobi  published  in  1829 
the  Fundamenta  nova  theoriae  functionum  ellipticarum, 
certain  of  the  results  of  which  had  been  simultane- 
ously discovered  by  Abel.  Legendre  had  recognised 
that  a  new  branch  of  analysis  was  involved  in  those 
investigations,  and  he  devoted  decades  of  earnest 
work  to  its  development.  Beginning  with  the  integral 
which  depends  upon  a  square  root  of  an  expression  of 

*  Cayley,  Address  to  the  British  Association,  etc.,  1883. 


ALGEBRA.  183 

the  fourth  degree  in  x,  Legendre  noticed  that  such 
integrals  can  be  reduced  to  canonical  forms.  At/r  = 
I/I — <£2sin2i/r  was  substituted  for  the  radical,  and 
three  essentially  different  classes  of  elliptic  integrals 
were  distinguished  and  represented  by  Fty),  £($}, 
II(</r).  These  classes  depend  upon  the  amplitude  i/r 
and  the  modulus  k,  the  last  class  also  upon  a  para- 
meter n. 

In  spite  of  the  elegant  investigations  of  Legendre 
on  elliptic  integrals,  their  theory  still  presented  sev 
eral  enigmatic  phenomena.  It  was  noticed  that  the 
degree  of  the  equation  conditioning  the  division  of 
the  elliptic  integral  is  not  equal  to  the  number  of  the 
parts,  as  in  the  division  of  the  circle,  but  to  its  square. 
The  solution  of  this  and  similar  problems  was  re- 
served for  Jacobi  and  Abel.  Of  the  many  productive 
ideas  of  these  two  eminent  mathematicians  there  are 
especially  two  which  belong  to  both  and  have  greatly 
advanced  the  theory. 

In  the  first  place,  Abel  and  Jacobi  independently  of 
each  other  observed  that  it  is  not  expedient  to  inves- 
tigate the  elliptic  integral  of  the  first  class  as  a  func- 
tion of  its  limits,  but  that  the  method  of  consideration 
must  be  reversed,  and  the  limit  introduced  as  a  func- 
tion of  two  quantities  dependent  upon  it.  Expressed 
in  other  words,  Abel  and  Jacobi  introduced  the  direct 
functions  instead  of  the  inverse.  Abel  called  them 
<£,  /,  F,  and  Jacobi  named  them  sin  am,  cos  am,  A  am, 
or,  as  they  are  written  by  Gudermann,  sn,  en,  dn. 


184  HISTORY  OF  MATHEMATICS. 

A  second  ingenious  idea,  which  belongs  to  Jacobi 
as  well  as  to  Abel,  is  the  introduction  of  the  imagi- 
nary into  this  theory.  As  Jacobi  himself  affirmed,  it 
was  just  this  innovation  which  rendered  possible  the 
solution  of  the  enigma  of  the  earlier  theory.  It  turned 
out  that  the  new  functions  partake  of  the  nature  of 
the  trigonometric  and  exponential  functions.  While 
the  former  are  periodic  only  for  real  values  of  the  ar- 
gument, and  the  latter  only  for  imaginary  values,  the 
elliptic  functions  have  two  periods.  It  can  safely  be 
said  that  Gauss  as  early  as  the  beginning  of  the  nine- 
teenth century  had  recognised  the  principle  of  the 
double  period,  a  fact  which  was  first  made  plain  in 
the  writings  of  Abel. 

Beginning  with  these  two  fundamental  ideas,  Ja- 
cobi and  Abel,  each  in  his  own  way,  made  further 
important  contributions  to  the  theory  of  elliptic  func- 
tions. Legendre  had  given  a  transformation  of  one 
elliptic  integral  into  another  of  the  same  form,  but  a 
second  transformation  discovered  by  him  was  un- 
known to  Jacobi,  as  the  latter  after  serious  difficulties 
reached  the  important  result  that  a  multiplication  in 
the  theory  of  such  functions  can  be  composed  of  two 
transformations.  Abel  applied  himself  to  problems 
concerning  the  division  and  multiplication  of  elliptic 
integrals.  A  thorough  study  of  double  periodicity  led 
him  to  the  discovery  that  the  general  division  of  the 
elliptic  integral  with  a  given  limit  is  always  algebraic- 
ally possible  as  soon  as  the  division  of  the  complete 


ALGEBRA.  185 

integrals  is  assumed  as  accomplished.  The  solution 
of  the  problem  was  applied  by  Abel  to  the  lemniscate, 
and  in  this  connection  it  was  proved  that  the  division 
of  the  whole  lemniscate  is  altogether  analogous  to 
that  of  the  circle,  and  can  be  performed  algebraically 
in  the  same  case.  Another  important  discovery  of 
Abel's  occurred  in  his  allowing,  for  elliptic  functions 
of  multiple  argument,  the  multiplier  to  become  infinite 
in  formulas  deduced  from  functions  with  a  single  ar- 
gument. From  this  resulted  the  remarkable  expres- 
sions which  represent  elliptic  functions  by  infinite 
series  or  quotients  of  infinite  products. 

Jacobi  had  assumed  in  his  investigations  on  trans- 
formations that  the  original  variable  is  rationally  ex- 
pressible in  terms  of  the  new.  Abel,  however,  entered 
this  field  with  the  more  general  assumption  that  be- 
tween these  two  quantities  an  algebraic  equation  ex- 
ists, and  the  result  of  his  labor  was  that  this  more 
general  problem  can  be  solved  by  the  help  of  the 
special  problem  completely  treated  by  Jacobi. 

Jacobi  carried  still  further  many  of  the  investiga- 
tions of  Abel.  Abel  had  given  the  theory  of  the  gen- 
eral division,  but  the  actual  application  demanded 
the  formation  of  certain  symmetric  functions  of  the 
roots  which  could  be  obtained  only  in  special  cases. 
Jacobi  gave  the  solution  of  the  problem  so  that  the 
required  functions  of  the  roots  could  be  obtained  at 
once  and  in  a  manner  simpler  than  Abel's.  When 
Jacobi  had  reached  this  goal,  he  stood  alone  on  the 


1 86  HISTORY  OF  MATHEMATICS. 

broad  expanse  of  the  new  science,  for  Abel  a  short 
time  before  had  found  an  early  grave  at  the  age  of  27. 

The  later  efforts  of  Jacobi  culminate  in  the  in- 
troduction of  the  theta-function.  Abel  had  already 
represented  elliptic  functions  as  quotients  of  infinite 
products.  Jacobi  could  represent  these  products  as 
special  cases  of  a  single  transcendent,  a  fact  which 
the  French  mathematicians  had  come  upon  in  physical 
researches  but  had  neglected  to  investigate.  Jacobi 
examined  their  analytic  nature,  brought  them  into 
connection  with  the  integrals  of  the  second  and  third 
class,  and  noticed  especially  that  integrals  of  the  third 
class,  though  dependent  upon  three  elements,  can  be 
represented  by  means  of  the  new  transcendent  involv- 
ing only  two  elements.  The  execution  of  this  process 
gave  to  the  whole  theory  a  high  degree  of  comprehen- 
siveness and  clearness,  allowing  the  elliptic  functions 
sn,  en,  dn  to  be  represented  with  the  new  Jacobian 
transcendents  ®i,  ©2,  ©3,  ©4  as  fractions  having  a  com- 
mon denominator. 

What  Abel  accomplished  in  the  theory  of  elliptic 
functions  is  conspicuous,  although  it  was  not  his 
greatest  achievement.  There  is  high  authority  for 
saying  that  the  achievements  of  Abel  were  as  great  in 
the  algebraic  field  as  in  that  of  elliptic  functions.  But 
his  most  brilliant  results  were  obtained  in  the  theory 
of  the  Abelian  functions  named  in  his  honor,  their 
first  development  falling  in  the  years  1826-1829. 
"Abel's  Theorem"  has  been  presented  by  its  discov- 


ALGEBRA.  I 87 

erer  in  different  forms.  The  paper,  Mtmoire  sur  une 
proprittt  gtntrale  d'une  classe  tres-ttendue  de  f one t ions 
transcendentes,  which  after  the  death  of  the  author  re- 
ceived the  prize  from  the  French  academy,  contained 
the  most  general  expression.  In  form  it  is  a  theorem 
of  the  integral  calculus,  the  integrals  depending  upon 
an  irrational  function  y,  which  is  connected  with  x  by 
an  algebraic  equation  F(x,  ^)=0.  Abel's  fundamental 
theorem  states  that  a  sum  of  such  integrals  can  be 
expressed  by  a  definite  number  /  of  similar  integrals 
where  p  depends  only  upon  the  properties  of  the  equa- 
tion F(x,  j>)=0.  (This/  is  the  deficiency  of  the  curve 
F(x,  ^)=0 ;  the  notion  of  deficiency,  however,  dates 
first  from  the  year  1857.)  For  the  case  that 


y  =  V  Ax*>  +  Bx*  +  Cx*  +  Dx+E, 
Abel's   theorem   leads   to   Legendre's   proposition   on 
the  sum  of  two  elliptic  integrals.      Here/  =  l.      If 


.  .  +  P, 

where  A  can  also  be  0,  then  p  is  2,  and  so  on.  For 
/>  =  S,  or  >  3,  the  hyperelliptic  integrals  are  only  spe- 
cial cases  of  the  Abelian  integrals  of  like  class. 

After  Abel's  death  (1829)  Jacobi  carried  the  theory 
further  in  his  Considerationes  generales  de  transcendenti- 
bus  Abelianis  (1832),  and  showed  for  hyperelliptic  in- 
tegrals of  a  given  class  that  the  direct  functions  to 
which  Abel's  proposition  applies  are  not  functions  of 
a  single  variable,  as  the  elliptic  functions  sn,  en,  dn, 
but  are  functions  of  /  variables.  Separate  papers  of 


1 88  HISTORY  OF   MArHEMATICS. 

essential  significance  for  the  case  /  =  2,  are  due  to 
Rosenhain  (1846,  published  1851)  and  Goepel  (1847). 

Two  articles  of  Riemann,  founded  upon  the  writ- 
ings of  Gauss  and  Cauchy,  have  become  significant 
in  the  development  of  the  complete  theory  of  func- 
tions. Cauchy  had  by  rigorous  methods  and  by  the 
introduction  of  the  imaginary  variable  "laid  the  foun- 
dation for  an  essential  improvement  and  transforma- 
tion of  the  whole  of  analysis."*  Riemann  built  upon 
this  foundation  and  wrote  the  Grundlage  fur  eine  all- 
gemeine  Theorie  der  Funktionen  einer  •veranderlichen 
komplexen  Grosse  in  the  year  1851,  and  the  Theorie  der 
AbeVschen  Funktionen  which  appeared  six  years  later. 
For  the  treatment  of  the  Abelian  functions,  Riemann 
used  theta-functions  with  several  arguments,  the  the- 
ory of  which  is  based  upon  the  general  principle  of 
the  theory  of  functions  of  a  complex  variable.  He 
begins  with  integrals  of  algebraic  functions  of  the 
most  general  form  and  considers  their  inverse  func- 
tions, that  is,  the  Abelian  functions  of  p  variables. 
Then  a  theta  function  of  /  variables  is  defined  as  the 
sum  of  a  /-tuply  infinite  exponential  series  whose 
general  term  depends,  in  addition  to  p  variables,  upon 
certain  — -  constants  which  must  be  reducible 

to  3/> — 3  moduli,  but  the  theory  has  not  yet  been  com- 
pleted. 

Starting  from  the  works  of  Gauss  and  Abel  as  well 

*  Kummer,  "  Gedachtnissrede  auf  Lejeune-Dirichlet,"  Berliner  Abhand- 
lungen,  1860. 


ALGEBRA.  189 

as  the  developments  of  Cauchy  on  integrations  in  the 
imaginary  plane,  a  strong  movement  appears  in  which 
occur  the  names  of  Weierstrass,  G.  Cantor,  Heine, 
Dedekind,  P.  Du  Bois-Reymond,  Dini,  Scheeffer, 
Pringsheim,  Holder,  Pincherle,  and  others.  This 
tendency  aims  at  freeing  from  criticism  the  founda- 
tions of  arithmetic,  especially  by  a  new  treatment  of 
irrationals  based  upon  the  theory  of  functions  with  its 
considerations  of  continuity  and  discontinuity.  It 
likewise  considers  the  bases  of  the  theory  of  series  by 
investigations  on  convergence  and  divergence,  and 
gives  to  the  differential  calculus  greater  preciseness 
through  the  introduction  of  mean-value  theorems. 

After  Riemann  valuable  contributions  to  the  theory 
of  the  theta-functions  were  made  by  Weierstrass, 
Weber,  Nother,  H.  Stahl,  Schottky,  and  Frobenius. 
Since  Riemann  a  theory  of  algebraic  functions  and 
point-groups  has  been  detached  from  the  theory  of 
Abelian  functions,  a  theory  which  was  founded  through 
the  writings  of  Brill,  Nother,  and  Lindemann  upon 
the  remainder-theorem  and  the  Riemann-Roch  theo- 
rem, while  recently  Weber  and  Dedekind  have  allied 
themselves  with  the  theory  of  ideal  numbers,  set  forth 
in  the  first  appendix  to  Dirichlet.  The  extremely 
rich  development  of  the  general  theory  of  functions 
in  recent  years  has  borne  fruit  in  different  branches  of 
mathematical  science,  and  undoubtedly  is  to  be  rec- 
ognised as  having  furnished  a  solid  foundation  for  the 
work  of  the  future. 


IV.  GEOMETRY. 

A.    GENERAL  SURVEY. 

THE  oldest  traces  of  geometry  are  found  among 
the  Egyptians  and  Babylonians.  In  this  first 
period  geometry  was  made  to  serve  practical  purposes 
almost  exclusively.  From  the  Egyptian  and  Baby- 
lonian priesthood  and  learned  classes  geometry  was 
transplanted  to  Grecian  soil.  Here  begins  the  second 
period,  a  classic  era  of  philosophic  conception  of  geo- 
metric notions  as  the  embodiment  of  a  general  science 
of  mathematics,  connected  with  the  names  of  Pythag- 
oras, Eratosthenes,  Euclid,  Apollonius,  and  Archi- 
medes. The  works  of  the  last  two  indeed,  touch  upon 
lines  not  clearly  defined  until  modern  times.  Apollo 
nius  in  his  Conic  Sections  gives  the  first  real  example 
of  a  geometry  of  position,  while  Archimedes  for  the 
most  part  concerns  himself  with  the  geometry  of  meas- 
urement. 

The  golden  age  of  Greek  geometry  was  brief  and 
yet  it  was  not  wholly  extinct  until  the  memory  of  the 
great  men  of  Alexandria  was  lost  in  the  insignificance 
of  their  successors.  Then  followed  more  than  a  thou- 


GEOMETRY.  IQI 

sand  years  of  a  cheerless  epoch  which  at  best  was  re- 
stricted to  borrowing  from  the  Greeks  such  geometric 
knowledge  as  could  be  understood.  History  might 
pass  over  these  many  centuries  in  silence  were  it  not 
compelled  to  give  attention  to  these  obscure  and  un- 
productive periods  in  their  relation  to  the  past  and 
future.  In  this  third  period  come  first  the  Romans, 
Hindus,  and  Chinese,  turning  the  Greek  geometry  to 
use  after  their  own  fashion ;  then  the  Arabs  as  skilled 
intermediaries  between  the  ancient  classic  and  a  mod- 
ern era. 

The  fourth  period  comprises  the  early  develop- 
ment of  geometry  among  the  nations  of  the  West. 
By  the  labors  of  Arab  authors  the  treasures  of  a  time 
long  past  were  brought  within  the  walls  of  monasteries 
and  into  the  hands  of  teachers  in  newly  established 
schools  and  universities,  without  as  yet  forming  a 
subject  for  general  instruction.  The  most  prominent 
intellects  of  this  period  are  Vieta  and  Kepler.  In 
their  methods  they  suggest  the  fifth  period  which  be- 
gins with  Descartes.  The  powerful  methods  of  analy- 
sis are  now  introduced  into  geometry.  Analytic  geom- 
etry comes  into  being.  The  application  of  its  seductive 
methods  received  the  almost  exclusive  attention  of 
the  mathematicians  of  the  seventeenth  and  eighteenth 
centuries.  Then  in  the  so-called  modern  or  projective 
geometry  and  the  geometry  of  curved  surfaces  there 
arose  theories  which,  like  analytic  geometry,  far  tran- 
scended the  geometry  of  the  ancients,  especially  in 


IQ2  HISTORY  OF  MATHEMATICS. 

the  way  of  leading  to  the  almost  unlimited  generaliza- 
tion of  truths  already  known. 


B.    FIRST  PERIOD. 

EGYPTIANS  AND  BABYLONIANS. 

In  the  same  book  of  Ahmes  which  has  disclosed  to 
us  the  elementary  arithmetic  of  the  Egyptians  are 
also  found  sections  on  geometry,  the  determination 
of  areas  of  simple  surfaces,  with  figures  appended. 
These  figures  are  either  rectilinear  or  circular.  Among 
them  are  found  isosceles  triangles,  rectangles,  isos- 
celes trapezoids  and  circles.*  The  area  of  the  rect- 
angle is  correctly  determined ;  as  the  measure  of  the 
area  of  the  isosceles  triangle  with  base  a  and  side  b, 
however,  \ab  is  found,  and  for  the  area  of  the  isosceles 
trapezoid  with  parallel  sides  a'  and  a"  and  oblique  side 
b,  the  expression  ^(a'-f-a")£  is  given.  These  approx- 
imate formulae  are  used  throughout  and  are  evidently 
considered  perfectly  correct.  The  area  of  the  circle 
follows,  with  the  exceptionally  accurate  value  ir  = 

=3.1605. 

Among  the  problems  of  geometric  construction 
one  stands  forth  preeminent  by  reason  of  its  practical 
importance,  viz.,  to  lay  off  a  right  angle.  The  solflv 
tion  of  this  problem,  so  vital  in  the  construction  of 
temples  and  palaces,  belonged  to  the  profession  of 

•  Cantor,  I.,  p.  52. 


GEOMETRY.  1 93 

rope- stretchers  or  harpedonaptae.  They  used  a  rope 
divided  by  knots  into  three  segments  (perhaps  corre- 
sponding to  the  numbers  3,  4,  5)  forming  a  Pythago- 
ean  triangle.* 

Among  the  Babylonians  the  construction  of  figures 
of  religious  significance  led  up  to  a  formal  geometry  of 
divination  which  recognized  triangles,  quadrilaterals, 
right  angles,  circles  with  the  inscribed  regular  hex- 
agon and  the  division  of  the  circumference  into  three 
hundred  and  sixty  degrees  as  well  as  a  value  ir  =  3. 

Stereometric  problems,  such  as  finding  the  con- 
tents of  granaries,  are  found  in  Ahmes;  but  not  much 
is  to  be  learned  from  his  statements  since  no  account 
is  given  of  the  shape  of  the  storehouses. 

As  for  projective  representations,  the  Egyptian 
wall-sculptures  show  no  evidence  of  any  knowledge 
of  perspective.  For  example  a  square  pond  is  pic- 
tured in  the  ground-plan  but  the  trees  and  the  water- 
drawers  standing  on  the  bank  are  added  to  the  picture 
in  the  elevation,  as  it  were  from  the  outside,  f 


C.    SECOND  PERIOD. 

THE  GREEKS. 

In  a  survey  of  Greek  geometry  it  will  here  and 
there  appear  as  if  investigations  connected  in  a  very 

*  Cantor,  I.,  p.  62. 

t  Wiener,  Lehrbuch  der  darstellenden  Geometric,  1884.    Hereafter  referred 
to  as  Wiener. 


IQ4  HISTORY  OF  MATHEMATICS. 

simple  manner  with  well-known  theorems  were  not 
known  to  the  Greeks.  At  least  it  seems  as  if  they 
could  not  have  been  established  satisfactorily,  since 
they  are  thrown  in  among  other  matters  evidently 
without  connection.  Doubtless  the  principal  reason 
for  this  is  that  a  number  of  the  important  writings  of 
the  ancient  mathematicians  are  lost.  Another  no  less 
weighty  reason  might  be  that  much  was  handed  down 
simply  by  oral  tradition,  and  the  latter,  by  reason  of 
the  stiff  and  repulsive  way  in  which  most  of  the  Greek 
demonstrations  were  worked  out,  did  not  always  ren- 
der the  truths  set  forth  indisputable. 

In  Thales  are  found  traces  of  Egyptian  geometry, 
but  one  must  not  expect  to  discover  there  all  that  was 
known  to  the  Egyptians.  Thales  mentions  the  theo- 
rems regarding  vertical  angles,  the  angles  at  the  base 
of  an  isosceles  triangle,  the  determination  of  a  triangle 
from  a  side  and  two  adjacent  angles,  and  the  angle  in- 
scribed in  a  semi-circle.  He  knew  how  to  determine 
the  height  of  an  object  by  comparing  its  shadow  with 
the  shadow  of  a  staff  placed  at  the  extremity  of  the 
shadow  of  the  object,  so  that  here  may  be  found  the 
beginnings  of  the  theory  of  similarity.  In  Thales  the 
proofs  of  the  theorems  are  either  not  given  at  all  or 
are  given  without  the  rigor  demanded  in  later  times. 

In  this  direction  an  important  advance  was  made 
by  Pythagoras  and  his  school.  To  him  without  ques- 
tion is  to  be  ascribed  the  theorem  known  to  the  Egyp- 
tian "rope-stretchers"  concerning  the  right-angled 


GEOMETRY.  1 95 

triangle,  which  they  knew  in  the  case  of  the  tri- 
angle with  sides  3,  4,  5,  without  giving  a  rigorous 
proof.  Euclid's  is  the  earliest  of  the  extant  proofs  of 
this  theorem.  Of  other  matters,  what  is  to  be  ascribed 
to  Pythagoras  himself,  and  what  to  his  pupils,  it  is 
difficult  to  decide.  The  Pythagoreans  proved  that  the 
sum  of  the  angles  of  a  plane  triangle  is  two  right  an- 
gles. They  knew  the  golden  section,  and  also  the 
regular  polygons  so  far  as  they  make  up  the  bound- 
aries of  the  five  regular  bodies.  Also  regular  star- 
polygons  were  known,  at  least  the  star-pentagon.  In 
the  Pythagorean  theorems  of  area  the  gnomon  played 
an  important  part.  This  word  originally  signified  the 
vertical  staff  which  by  its  shadow  indicated  the  hours, 
and  later  the  right  angle  mechanically  represented. 
Among  the  Pythagoreans  the  gnomon  is  the  figure 
left  after  a  square  has  been  taken  from  the  corner  of 
another  square.  Later,  in  Euclid,  the  gnomon  is  a 
parallelogram  after  similar  treatment  (see  page  66). 
The  Pythagoreans  called  the  perpendicular  to  a  straight 
line  "a  line  directed  according  to  the  gnomon."* 

But  geometric  knowledge  extended  beyond  the 
school  of  Pythagoras.  Anaxagoras  is  said  to  have  been 
the  first  to  try  to  determine  a  square  of  area  equal 
to  that  of  a  given  circle.  It  is  to  be  noticed  that  like 
most  of  his  successors  he  believed  in  the  possibility 
of  solving  this  problem.  OEnopides  showed  how  to 
draw  a  perpendicular  from  a  point  to  a  line  and  how 

*  Cantor,  I.,  p.  150. 


i96 


HISTORY  OF  MATHEMATICS. 


to  lay  off  a  given  angle  at  a  given  point  of  a  given 
line.  Hippias  of  Elis  likewise  sought  the  quadrature 
of  the  circle,  and  later  he  attempted  the  trisection  of 
an  angle,  for  which  he  constructed  the  quadratrix. 

B 


This  curve  is  described  as  follows  :  Upon  a  quadrant  of  a  cir- 
cumference cut  off  by  two  perpendicular  radii,  OA  and  OB,  lie 
the  points  A,  ...  1C,  L,  ...  B.  The  radius  r=  OA  revolves  with 
uniform  velocity  about  O  from  the  position  OA  to  the  position  OB. 
At  the  same  time  a  straight  line  g  always  parallel  to  OA  moves 
with  uniform  velocity  from  the  position  OA  to  that  of  a  tangent  to 
the  circle  at  B.  If  K'  is  the  intersection  of  g  with  OB  at  the  time 
when  the  moving  radius  falls  upon  OJTtben  the  parallel  to  OA 
through  K'  meets  the  radius  OJf'm  a  point  K"  belonging  to  the 
quadratrix.  If  P  is  the  intersection  of  OA  with  the  quadratrix,  it 
follows  in  part  directly  and  in  part  from  simple  considerations,  that 

arc  AK  =  OK' 
arc  AL  ~   OL'  ' 

a.  relation  which  solves  any  problem  of  angle  sections.     Further- 
more, 

1r         OP  OA 


- 

~  w '        OA  ~  arc  Aff 

whence  it  is  obvious  that  the  quadrature  of  the  circle  depends  upon 


GEOMETRY.  197 

the  ratio  in  which  the  radius  OA  is  divided  by  the  point  P  of  the 
quadratrix.  If  this  ratio  could  be  constructed  by  elementary  geom- 
etry, the  quadrature  of  the  circle  would  be  effected.*  It  appears 
that  the  quadratrix  was  first  invented  for  the  trisection  of  an  angle 
and  that  its  relation  to  the  quadrature  of  the  circle  was  discovered 
later,  f  as  is  shown  by  Dinostratus. 

The  problem  of  the  quadrature  of  the  circle  is  also 
found  in  Hippocrates.  He  endeavored  to  accomplish 
his  purpose  by'the  consideration  of  crescent-shaped 
figures  bounded  by  arcs  of  circles.  It  is  of  especial 
importance  to  note  that  Hippocrates  wrote  an  ele- 
mentary book  of  mathematics  (the  first  of  the  kind) 
in  which  he  represented  a  point  by  a  single  capital 
letter  and  a  segment  by  tuo,  although  we  are  unable 
to  determine  who  was  the  first  to  introduce  this  sym- 
bolism. £ 

Geometry  was  strengthened  on  the  philosophic 
side  by  Plato,  who  felt  the  need  of  establishing  defini- 
tions and  axioms  and  simplifying  the  work  of  the  in- 
vestigator by  the  introduction  of  the  analytic  method. 

A  systematic  representation  of  the  results  of  all 
the  earlier  investigations  in  the  domain  of  elementary 
geometry,  enriched  by  the  fruits  of  his  own  abundant 
labor,  is  given  by  Euclid  in  the  thirteen  books  of  his 
Elements  which  deal  not  only  with  plane  figures  but 
also  with  figures  in  space  and  algebraic  investiga- 

*The  equation  of  the  quadratrix  in  polar  co-ordinates  is  r=  —  .  ,"  , 
where  a  =  OA.  Putting  <£=o,  r  =  ra,  we  have  ir=  — . 


198  HISTORY  OF  MATHEMATICS. 

tions.  "Whatever  has  been  said  in  praise  of  mathe- 
matics, of  the  strength,  perspicuity  and  rigor  of  its 
presentation,  all  is  especially  true  of  this  work  of  the 
great  Alexandrian.  Definitions,  axioms,  and  conclu- 
sions are  joined  together  link  by  link  as  into  a  chain, 
firm  and  inflexible,  of  binding  force  but  also  cold  and 
hard,  repellent  to  a  productive  mind  and  affording  no 
room  for  independent  activity.  A  ripened  understand- 
ing is  needed  to  appreciate  the  classic  beauties  of  this 
greatest  monument  of  Greek  ingenuity.  It  is  not  the 
arena  for  the  youth  eager  for  enterprise ;  to  captivate 
him  a  field  of  action  is  better  suited  where  he  may 
hope  to  discover  something  new,  unexpected."* 

The  first  book  of  the  Elements  deals  with  the  the- 
ory of  triangles  and  quadrilaterals,  the  second  book 
with  the  application  of  the  Pythagorean  theorem  to 
a  large  number  of  constructions,  really  of  arithmetic 
nature.  The  third  book  introduces  circles,  the  fourth 
book  inscribed  and  circumscribed  polygons.  Propor- 
tions explained  by  the  aid  of  line-segments  occupy 
the  fifth  book,  and  in  the  sixth  book  find  their  appli- 
cation to  the  proof  of  theorems  involving  the  similar- 
ity of  figures.  The  seventh,  eighth,  ninth  and  tenth 
books  have  especially  to  do  with  the  theory  of  num- 
bers. These  books  contain  respectively  the  measure- 
ment and  division  of  numbers,  the  algorism  for  de- 
termining the  least  common  multiple  and  the  greatest 
common  divisor,  prime  numbers,  geometric  series, 

*A.  Brill,  Antrittsrede  in  Tubingen,  1884. 


GEOMETRY.  IQ9 

and  incommensurable  (irrational)  numbers.  Then 
follows  stereometry :  in  the  eleventh  book  the  straight 
line,  the  plane,  the  prism  ;  in  the  twelfth,  the  discus- 
sion of  the  prism,  pyramid,  cone,  cylinder,  sphere; 
and  in  the  thirteenth,  regular  polygons  with  the  regu- 
lar solids  formed  from  them,  the  number  of  which 
Euclid  gives  definitely  as  five.  Without  detracting  in 
the  least  from  the  glory  due  to  Euclid  for  the  compo- 
sition of  this  imperishable  work,  it  may  be  assumed 
that  individual  portions  grew  out  of  the  well  grounded 
preparatory  work  of  others.  This  is  almost  certainly 
true  of  the  fifth  book,  of  which  Eudoxus  seems  to 
have  been  the  real  author. 

Not  by  reason  of  a  great  compilation  like  Euclid, 
but  through  a  series  of  valuable  single  treatises,  Archi- 
medes is  justly  entitled  to  have  a  more  detailed  de- 
scription of  his  geometric  productions.  In  his  inves- 
tigations of  the  sphere  and  cylinder  he  assumes  that 
the  straight  line  is  the  shortest  distance  between  two 
points.  From  the  Arabic  we  have  a  small  geometric 
work  of  Archimedes  consisting  of  fifteen  so-called 
lemmas,  some  of  which  have  value  in  connection  with 
the  comparison  of  figures  bounded  by  straight  lines 
and  arcs  of  circles,  the  trisection  of  the  angle,  and 
the  determination  of  cross-ratios.  Of  especial  impor- 
tance is  his  mensuration  of  the  circle,  in  which  he 
shows  IT  to  lie  between  3^  and  3|£.  This  as  well  as 
many  other  results  Archimedes  obtains  by  the  method 
of  exhaustions  which  among  the  ancients  usually  took 


2OO  HISTORY  OF  MATHEMATICS. 

the  place  of  the  modern  integration.*  The  quantity 
sought,  the  area  bounded  by  a  curve,  for  example, 
may  be  considered  as  the  limit  of  the  areas  of  the  in- 
scribed and  circumscribed  polygons  the  number  of 
whose  sides  is  continually  increased  by  the  bisection 
of  the  arcs,  and  it  is  shown  that  the  difference  between 
two  associated  polygons,  by  an  indefinite  continuance 
of  this  process,  must  become  less  than  an  arbitrarily 
small  given  magnitude.  This  difference  was  thus,  as 
it  were,  exhausted,  and  the  result  obtained  by  exhaus- 
tion. 

The  field  of  the  constructions  of  elementary  geom- 
etry received  at  the  hands  of  Apollonius  an  extension 
in  the  solution  of  the  problem  to  construct  a  circle 
tangent  to  three  given  circles,  and  in  the  systematic 
introduction  of  the  diorismus  (determination  or  limi- 
tation). This  also  appears  in  more  difficult  problems 
in  his  Conic  Sections,  from  which  we  see  that  Apollo- 
nius gives  not  simply  the  conditions  for  the  possibility 
of  the  solution  in  general,  but  especially  desires  to 
determine  the  limits  of  the  solutions. 

From  Zenodorus  several  theorems  regarding  iso- 
perimetric  figures  are  still  extant  ;  for  example,  he 
states  that  the  circle  has  a  greater  area  than  any  iso- 
perimetric  regular  polygon,  that  among  all  isoperi- 
metric  polygons  of  the  same  number  of  sides  the  reg- 
ular has  the  greatest  area,  and  so  on.  Hypsicles  gives 


*Chasles,  Afersu  historigue  sur  I'orfgine  et  It  dtveloppement  des  mtthode 
en  giomltrie,  1875.     Hereafter  referred  to  as  Chasles. 


GEOMETRY.  2OI 

as  something  new  the  division  of  the  circumference 
into  three  hundred  and  sixty  degrees.  From  Heron 
we  have  a  book  on  geometry  (according  to  Tannery 
still  another,  a  commentary  on  Euclid's  Elements) 
which  deals  in  an  extended  manner  with  the  mensu- 
ration of  plane  figures.  Here  we  find  deduced  for  the 
area  A  of  the  triangle  whose  sides  are  a,  b,  and  c, 
where  2s=a-\-b  -\-  c,  the  formula 


In  the  measurement  of  the  circle  we  usually  find  %f.  as 
an  approximation  for  ir;  but  still  in  the  Book  of  Meas- 
urements we  also  find  TT  =  3. 

In  the  period  after  the  commencement  of  the 
Christian  era  the  output  becomes  still  more  meager. 
Only  occasionally  do  we  find  anything  noteworthy. 
Serenus,  however,  gives  a  theorem  on  transversals 
which  expresses  the  fact  that  a  harmonic  pencil  is  cut 
by  an  arbitrary  transversal  in  a  harmonic  range.  In 
the  Almagest  occurs  the  theorem  regarding  the  in- 
scribed quadrilateral,  ordinarily  known  as  Ptolemy's 
Theorem,  and  a  value  written  in  sexagesimal  form 
7r  =  3.8.30,  i.  e., 

*  =  "  +  TO  +  60-60  =  3  Jl  =  3.14166....* 

In  a  special  treatise  on  geometry  Ptolemy  shows  that 
he  does  not  regard  Euclid's  theory  of  parallels  as  in- 
disputable. 

*  Cantor,  I.,  p.  394. 


202  HISTORY  OF  MATHEMATICS. 

To  the  last  supporters  of  Greek  geometry  belong 
Sextus  Julius  Africanus,  who  determined  the  width  of 
a  stream  by  the  use  of  similar  right-angled  triangles, 
and  Pappus,  whose  name  has  become  very  well  known 
by  reason  of  his  Collection.  This  work  consisting  orig- 
inally of  eight  books,  of  which  the  first  is  wholly 
lost  and  the  second  in  great  part,  presents  the  sub- 
stance of  the  mathematical  writings  of  special  repute 
in  the  time  of  the  author,  and  in  some  places  adds 
corollaries.  Since  his  work  was  evidently  composed 
with  great  conscientiousness,  it  has  become  one  of 
the  most  trustworthy  sources  for  the  study  of  the 
mathematical  history  of  ancient  times.  The  geomet- 
ric part  of  the  Collection  contains  among  other  things 
discussions  of  the  three  different  means  between  two 
line-segments,  isoperimetric  figures,  and  tangency  of 
circles.  It  also  discusses  similarity  in  the  case  of  cir- 
cles ;  so  far  at  least  as  to  show  that  all  lines  which 
join  the  ends  of  parallel  radii  of  two  circles,  drawn  in 
the  same  or  in  opposite  directions,  intersect  in  a  fixed 
point  of  the  line  of  centers. 

The  Greeks  rendered  important  service  not  simply 
in  the  field  of  elementary  geometry :  they  are  also  the 
creators  of  the  theory  of  conic  sections.  And  as  in 
the  one  the  name  of  Euclid,  so  in  the  other  the  name 
of  Apollonius  of  Perga  has  been  the  signal  for  con- 
troversy. The  theory  of  the  curves  of  second  order 
does  not  begin  with  Apollonius  any  more  than  does 
Euclidean  geometry  begin  with  Euclid ;  but  what  the 


GEOMETRY.  2O3 

Elements  signify  for  elementary  geometry,  the  eight 
books  of  the  Conies  signify  for  the  theory  of  lines  of 
the  second  order.  Only  the  first  four  books  of  the 
Conic  Sections  of  Apollonius  are  preserved  in  the 
Greek  text :  the  next  three  are  known  through  Arabic 
translations :  the  eighth  book  has  never  been  found 
and  is  given  up  for  lost,  though  its  contents  have  been 
restored  by  Halley  from  references  in  Pappus.  The 
first  book  deals  with  the  formation  of  conies  by  plane 
sections  of  circular  cones,  with  conjugate  diameters, 
and  with  axes  and  tangents.  The  second  has  espe- 
cially to  do  with  asymptotes.  These  Apollonius  ob- 
tains by  laying  off  on  a  tangent  from  the  point  of  con- 
tact the  half-length  of  the  parallel  diameter  and  joining 
its  extremity  to  the  center  of  the  curve.  The  third 
book  contains  theorems  on  foci  and  secants,  and  the 
fourth  upon  the  intersection  of  circles  with  conies  and 
of  conies  with  one  another.  With  this  the  elementary 
treatment  of  conies  by  Apollonius  closes.  The  fol- 
lowing books  contain  special  investigations  in  applica- 
tion of  the  methods  developed  in  the  first  four  books. 
Thus  the  fifth  book  deals  with  the  maximum  and  min- 
imum lines  which  can  be  drawn  from  a  point  to  the 
conic,  and  also  with  the  normals  from  a  given  point 
in  the  plane  of  the  curve  of  the  second  order;  the  sixth 
with  equal  and  similar  conies ;  the  seventh  in  a  re- 
markable manner  with  the  parallelograms  having  con- 
jugate diameters  as  sides  and  the  theorem  upon  the 
sum  of  the  squares  of  conjugate  diameters.  The  eighth 


204  HISTORY  OF  MATHEMATICS. 

book  contained,  according  to  Halley,  a  series  of  prob- 
lems connected  in  the  closest  manner  with  lemmas  of 
the  seventh  book. 

The  first  effort  toward  the  development  of  the  the- 
ory of  conic  sections  is  ascribed  to  Hippocrates.*  He 
reduced  the  duplication  of  the  cube  to  the  construc- 
tion of  two  mean  proportionals  x  and  ^  between  two 
given  line-segments  a  and  b ;  thusf 

—  =  —  =  y  gives  x1*  =  ay,  y*  =  bx,  whence 

Xs  =  a^b  •=•  —  a3  =  m  •  a3. 
a 

Archytas  and  Eudoxus  seem  to  have  found,  by  plane 
construction,  curves  satisfying  the  above  equations 
but  different  from  straight  lines  and  circles.  Menaech- 
mus  sought  for  the  new  curves,  already  known  by 
plane  constructions,  a  representation  by  sections  of 
cones  of  revolution,  and  became  the  discoverer  of 
conic  sections  in  this  sense.  He  employed  only  sec- 
tions perpendicular  to  an  element  of  a  right  circular 
cone;  thus  the  parabola  was  designated  as  the  "sec- 
tion of  a  right-angled  cone"  (whose  generating  angle 
is  45°) ;  the  ellipse,  the  "  section  of  an  acute-angled 
cone";  the  hyperbola,  the  "section  of  an  obtuse- 
angled  cone."  These  names  are  also  used  by  Archi- 
medes, although  he  was  aware  that  the  three  curves 
can  be  formed  as  sections  of  any  circular  cone.  Apol- 

"Zeuthen,  Die  Lehre  von  den  Kegelschnitten  im  Alter  turn.    Deutsch  von 
v.  Fischer-Benzon,  1886.    P.  459.    Hereafter  referred  to  as  Zeuthen. 
t  Cantor,  I.,  p.  200. 


GEOMETRY.  2O5 

lonius  first  introduced  the  names  "ellipse,'  "para- 
bola," "hyperbola."  Possibly  Menaechmus,  but  in 
any  case  Archimedes,  determined  conies  by  a  linear 
equation  between  areas,  of  the  form  y*  =  kxxi.  The 
semi-parameter,  with  Archimedes  and  possibly  some 
of  his  predecessors,  was  known  as  "the  segment  to 
the  axis,"  i.  e.,  the  segment  of  the  axis  of  the  circle 
from  the  vertex  of  the  curve  to  its  intersection  with 
the  axis  of  the  cone.  The  designation  "parameter" 
is  due  to  Desargues  (1639).* 

It  has  been  shown  f  that  Apollonius  represented  the  conies  by 
equations  of  the  form  y2=px-^-ax2,  where  x  and  y  are  regarded 
as  parallel  coordinates  and  every  term  is  represented  as  an  area. 
From  this  other  linear  equations  involving  areas  were  derived,  and 
so  equations  belonging  to  analytic  geometry  were  obtained  by  the 
use  of  a  system  of  parallel  coordinates  whose  origin  could,  for 
geometric  reasons,  be  shifted  simultaneously  with  an  interchange 
of  axes.  Hence  we  already  find  certain  fundamental  ideas  of  the 
analytic  geometry  which  appeared  almost  two  thousand  years  later. 

The  study  of  conic  sections  was  continued  upon  the 
cone  itself  only  till  the  time  when  a  single  fundamen- 
tal plane  property  rendered  it  possible  to  undertake 
the  further  investigation  in  the  plane.  J  In  this  way 
there  had  become  known,  up  to  the  time  of  Archi- 
medes, a  number  of  important  theorems  on  conjugate 
diameters,  and  the  relations  of  the  lines  to  these  di- 
ameters as  axes,  by  the  aid  of  linear  equations  be- 

*Baltzer,  R.,  Analytische  Geometrfe,  1882. 
tZeuthen,  p.  32.  tZeuthen,  p.  43. 


2O6  HISTORY  OF  MATHEMATICS*. 

tween  areas.  There  were  also  known  the  so-called 
Newton's  power-theorem,  the  theorem  that  the  rect- 
angles of  the  segments  of  two  secants  of  a  conic  drawn 
through  an  arbitrary  point  in  given  direction  are  in  a 
constant  ratio,  theorems  upon  the  generation  of  a 
conic  by  aid  of  its  tangents  or  as  the  locus  related  to 
four  straight  lines,  and  the  theorem  regarding  pole 
and  polar.  But  these  theorems  were  always  applied 
to  only  one  branch  of  the  hyperbola.  One  of  the  valu- 
able services  of  Apollonius  was  to  extend  his  own 
theorems,  and  consequently  those  already  known,  to 
both  branches  of  the  hyperbola.  His  whole  method 
justifies  us  in  regarding  him  the  most  prominent  rep- 
resentative of  the  Greek  theory  of  conic  sections,  and 
so  much  the  more  when  we  can  see  from  his  principal 
work  that  the  foundations  for  the  theory  of  projective 
ranges  and  pencils  had  virtually  been  laid  by  the  an- 
cients in  different  theorems  and  applications. 

With  Apollonius  the  period  of  new  discoveries  in 
the  realm  of  the  theory  of  conies  comes  to  an  end.  In 
later  times  we  find  only  applications  of  long  known 
theorems  to  problems  of  no  great  difficulty.  Indeed, 
the  solution  of  problems  already  played  an  important 
part  in  the  oldest  times  of  Greek  geometry  and  fur- 
nished the  occasion  for  the  exposition  not  only  of 
conies  but  also  of  curves  of  higher  order  than  the  sec- 
ond. In  the  number  of  problems,  which  on  account 
of  their  classic  value  have  been  transmitted  from  gen- 
eration to  generation  and  have  continually  furnished 


GEOMETRY.  2O7 

occasion  for  further  investigation,  three,  by  reason  of 
their  importance,  stand  forth  preeminent :  the  duplica- 
tion of  the  cube,  or  more  generally  the  multiplication 
of  the  cube,  the  trisection  of  the  angle  and  the  quad- 
rature of  the  circle.  The  appearance  of  these  three 
problems  has  been  of  the  greatest  significance  in  the 
development  of  the  whole  of  mathematics.  The  first 
requires  the  solution  of  an  equation  of  the  third  de- 
gree ;  the  second  (for  certain  angles  at  least)  leads  to 
an  important  section  of  the  theory  of  numbers,  i.  e., 
to  the  cyclotomic  equations,  and  Gauss  (see  p.  160) 
was  the  first  to  show  that  by  a  finite  number  of  ope- 
rations with  straight  edge  and  compasses  we  can  con- 
struct a  regular  polygon  of  n  sides  only  when  n — 1 
=  2a/  (p  an  arbitrary  integer).  The  third  problem 
reaches  over  into  the  province  of  algebra,  for  Linde- 
mann*  in  the  year  1882  showed  that  ir  cannot  be  the 
root  of  an  algebraic  equation  with  integral  coefficients. 
The  multiplication  of  the  cube,  algebraically  the 
determination  of  x  from  the  equation 

x3  =  —  •a*=m-a*, 
a 

is  also  called  the  Delian  problem,  because  the  Delians 
were  required  to  double  their  cubical  altar,  f  The  so- 
lution of  this  problem  was  specially  studied  by  Plato, 
Archytas,  and  Menaechmus;  the  latter  solved  it  by 

*Mathem.  Annalen,  XX.,  p.  215.  See  also  Mathem.  Annalen,  XLIII.,  and 
Klein,  Famous  Problems  of  Elementary  Geometry,  1895,  translated  by  Beman 
and  Smith,  Boston,  1897. 

t  Cantor,  I.,  p.  219. 


2O8  HISTORY  OF  MATHEMATICS. 

the  use  of  conies  (hyperbolas  and  parabolas).  Era- 
tosthenes constructed  a  mechanical  apparatus  for  the 
same  purpose. 

Among  the  solutions  of  the  problem  of  the  trisec- 
tion  of  an  angle,  the  method  of  Archimedes  is  note- 
worthy. It  furnishes  an  example  of  the  so-called 
"insertions"  of  which  the  Greeks  made  use  when  a 
solution  by  straight  edge  and  compasses  was  impos- 
sible. His  process  was  as  follows :  Required  to  divide 
the  arc  AB  of  the  circle  with  center  M  into  three 
equal  parts.  Draw  the  diameter  AE,  and  through  B 
a  secant  cutting  the  circumference  in  C  and  the  di- 
ameter AE  in  D,  so  that  CD  equals  the  radius  r  of 
the  circle.  Then  arc  CE  =  \AB. 


According  to  the  rules  of  insertion  the  process  con- 
sists in  laying  off  upon  a  ruler  a  length  r,  causing  it 
to  pass  through  B  while  one  extremity  D  of  the  seg- 
ment r  slides  along  the  diameter  AE.  By  moving 
the  ruler  we  get  a  certain  position  in  which  the  other 
extremity  of  the  segment  r  falls  upon  the  circumfer- 
ence, and  thus  the  point  C  is  determined. 

This  problem  Pappus  claims  to  have  solved  after 
the  manner  of  the  ancients  by  the  use  of  conic  sec- 


GEOMETRY.  209 

tions.  Since  in  the  writings  of  Apollonius,  so  largely 
lost,  lines  of  the  second  order  find  an  extended  appli- 
cation to  the  solution  of  problems,  the  conies  were 
frequently  called  solid  loci  in  opposition  to  plane  loci, 
i.  e.,  the  straight  line  and  circle.  Following  these 
came  linear  loci,  a  term  including  all  other  curves,  of 
which  a  large  number  were  investigated. 

This  designation  of  the  loci  is  found,  for  example, 
in  Pappus,  who  says  in  his  seventh  book*  that  a  prob- 
lem is  called  plane,  solid,  or  linear,  according  as  its 
solution  requires  plane,  solid,  or  linear  loci.  It  is, 
however,  highly  probable  that  the  loci  received  their 
names  from  problems,  and  that  therefore  the  division 
of  problems  into  plane,  solid,  and  linear  preceded  the 
designation  of  the  corresponding  loci.  First  it  is  to 
be  noticed  that  we  do  not  hear  of  "linear  problems 
and  loci"  till  after  the  terms  "plane  and  solid  prob- 
lems and  loci"  were  in  use.  Plane  problems  were 
those  which  in  the  geometric  treatment  proved  to  be 
dependent  upon  equations  of  the  first  or  second  de- 
gree between  segments,  and  hence  could  be  solved 
by  the  simple  application  of  areas,  the  Greek  method 
for  the  solution  of  quadratic  equations.  Problems  de- 
pending upon  the  solution  of  equations  of  the  third 
degree  between  segments  led  to  the  use  of  forms  of 
three  dimensions,  as,  e.  g.,  the  duplication  of  the 
cube,  and  were  termed  solid  problems;  the  loci  used 
in  their  solution  (the  conies)  were  solid  loci.  At  a 

*Zeuthen,  p.  203. 


2IO  HISTORY  OF  MATHEMATICS. 

time  when  the  significance  of  "plane"  and  "solid" 
was  forgotten,  the  term  "linear  problem"  was  first 
applied  to  those  problems  whose  treatment  (by  "lin- 
ear loci")  no  longer  led  to  equations  of  the  first,  sec- 
ond, and  third  degrees,  and  which  therefore  could  no 
longer  be  represented  as  linear  relations  between  seg- 
ments, areas,  or  volumes. 

Of  linear  loci  Hippias  applied  the  quadratrix  (to 
which  the  name  of  Dinostratus  was  later  attached 
through  his  attempt  at  the  quadrature  of  the  circle)* 
to  the  trisection  of  the  angle.  Eudoxus  was  acquainted 
with  the  sections  of  the  torus  made  by  planes  parallel 
to  the  axis  of  the  surface,  especially  the  hippopede  or 
figure-of-eight  curve,  f  The  spirals  of  Archimedes 
attained  special  celebrity.  His  exposition  of  their 
properties  compares  favorably  with  his  elegant  inves- 
tigations of  the  quadrature  of  the  parabola. 

Conon  had  already  generated  the  spiral  of  Archi- 
medes J  by  the  motion  of  a  point  which  recedes  with 
uniform  velocity  along  the  radius  OA  of  a  circle  k 
from  the  center  O,  while  OA  likewise  revolves  uni- 
formly about  O.  But  Archimedes  was  the  first  to  dis- 
cover certain  of  the  beautiful  properties  of  this  curve; 
he  found  that  if,  after  one  revolution,  the  spiral  meets 
the  circle  k  of  radius  OA  in  B  (where  BO  is  tangent 
to  the  spiral  at  O),  the  area  bounded  by  BO  and  the 

*  Cantor,  I.,  pp.  184,  233. 

t  Majer,  Proklo*  iiber  die  Petita  und  Axiomata  bet  Evklid,  1875. 

t  Cantor,  I.,  p.  291. 


GEOMETRY.  211 

spiral  is  one-third  of  the  area  of  the  circle  k\  further 
that  the  tangent  to  the  spiral  at  B  cuts  off  from  a  per- 
pendicular to  OB  at  O  a  segment  equal  to  the  circum- 
ference of  the  circle  k.* 

The  only  noteworthy  discovery  of  Nicomedes  is 
the  construction  of  the  conchoid  which  he  employed 
to  solve  the  problem  of  the  two  mean  proportionals, 
or,  what  amounts  to  the  same  thing,  the  multiplica- 
tion of  the  cube.  The  curve  is  the  geometric  locus  of 
the  point  X  upon  a  moving  straight  line  g  which  con- 
stantly passes  through  a  fixed  point  P  and  cuts  a  fixed 
straight  line  h  in  Fso  that  XY  has  a  constant  length. 
Nicomedes  also  investigated  the  properties  of  this 
curve  and  constructed  an  apparatus  made  of  rulers 
for  its  mechanical  description. 

The  cissoid  of  Diocles  is  also  of  use  in  the  multi- 
plication of  the  cube.  It  may  be  constructed  as  fol- 
lows :  Through  the  extremity  A  of  the  radius  OA  of 
a  circle  k  passes  the  secant  AC  which  cuts  k  in  C  and 
the  radius  OB  perpendicular  to  OA  in  D\  X,  upon 
AC,  is  a  point  of  the  cissoid  when  DX=DC.\  Gemi- 
nus  proves  that  besides  the  straight  line  and  the  circle 
the  common  helix  invented  by  Archytas  possesses  the 
insertion  property. 

Along  with  the  geometry  of  the  plane  was  devel- 
oped the  geometry  of  space,  first  as  elementary  stere- 


*Montucla. 

+  Klein,  Y.,  Famous  Problems  of  Elementary  Geometry,  translated  by  Beman 
ind  Smith,  Boston,  1897,  p.  44. 


212  HISTORY  OF  MATHEMATICS. 

ometry  and  then  in  theorems  dealing  with  surfaces  of 
the  second  order.  The  knowledge  of  the  five  regular 
bodies  and  the  related  circumscribed  sphere  certainly 
goes  back  to  Pythagoras.  According  to  the  statement 
of  Timaeus  of  Locri,*  fire  is  made  up-of  tetrahedra, 
air  of  octahedra,  water  of  icosahedra,  earth  of  cubes, 
while  the  dodecahedron  forms  the  boundary  of  the 
universe.  Of  these  five  cosmic  or  Platonic  bodies 
Theaetetus  seems  to  have  been  the  first  to  publish  a 
connected  treatment.  Eudoxus  states  that  a  pyramid 
(or  cone)  is  £  of  a  prism  of  equal  base  and  altitude. 
The  eleventh,  twelfth  and  thirteenth  books  of  Euclid's 
Elements  offer  a  summary  discussion  of  the  ordinary 
stereometry.  (See  p.  199.)  Archimedes  introduces 
thirteen  semi-regular  solids,  i.  e.,  solids  whose  bound- 
aries are  regular  polygons  of  two  or  three  different 
kinds.  Besides  this  he  compares  the  surface  and  vol- 
ume of  the  sphere  with  the  corresponding  expressions 
for  the  circumscribed  cylinder  and  deduces  theorems 
which  he  esteems  so  highly  that  he  expresses  the  de- 
sire to  have  the  sphere  and  circumscribed  cylinder 
cut  upon  his  tomb-stone.  Among  later  mathemati- 
cians Hypsicles  and  Heron  give  exercises  in  the  men- 
suration of  regular  and  irregular  solids.  Pappus  also 
furnishes  certain  stereometric  investigations  of  which 
we  specially  mention  as  new  only  the  determination 
of  the  volume  of  a  solid  of  revolution  by  means  of  the 
meridian  section  and  the  path  of  its  center  of  gravity. 

*  Cantor,  I.,  p.  163. 


GEOMETRY.  213 

He  thus  shows  familiarity  with  a  part  of  the  theorem 
later  known  as  Guldin's  rule. 

Of  surfaces  of  the  second  order  the  Greeks  knew 
the  elementary  surfaces  of  revolution,  i.  e.,  the  sphere, 
the  right  circular  cylinder  and  circular  cone.  Euclid 
deals  only  with  cones  of  revolution,  Archimedes  on  the 
contrary  with  circular  cones  in  general.  In  addition, 
Archimedes  investigates  the  "right-angled  conoids" 
(paraboloids  of  revolution),  the  "obtuse-angled  co- 
noids" (hyperboloids  of  revolution  of  one  sheet),  and 
"long  and  flat  spheroids"  (ellipsoids  of  revolution 
about  the  major  and  minor  axes).  He  determines  the 
character  of  plane  sections  and  the  volume  of  seg- 
ments of  such  surfaces.  Probably  Archimedes  also 
knew  that  these  surfaces  form  the  geometric  locus  of 
a  point  whose  distances  from  a  fixed  point  and  a  given 
plane  are  in  a  constant  ratio.  According  to  Proclus,* 
who  is  of  importance  as  a  commentator  upon  Euclid, 
the  torus  was  also  known — a  surface  generated  by  a 
circle  of  radius  r  revolving  about  an  axis  in  its  plane 
so  that  its  center  describes  a  circle  of  radius  <?.  The 
cases  ^  =  ^,  >  e,  <<?  were  discussed. 

With  methods  of  projection,  also,  the  Greeks  were 
not  unacquainted.f  Anaxagoras  and  Democritus  are 
said  to  have  known  the  laws  of  the  vanishing  point 
and  of  reduction,  at  least  for  the  simplest  cases.  Hip- 
parchus  projects  the  celestial  sphere  from  a  pole  upon 

*Majer,  Proklos  iiber  die  Petita  und  Axiomata  bet  Euklid,  1875. 
t  Wiener. 


214  HISTORY  OF  MATHEMATICS. 

the  plane  of  the  equator;  he  is  therefore  the  inventor 
of  the  stereographic  projection  which  has  come  to  be 
known  by  the  name  of  Ptolemy. 


D.   THIRD  PERIOD. 

ROMANS,   HINDUS,   CHINESE,   ARABS. 

Among  no  other  people  of  antiquity  did  geometry 
reach  so  high  an  eminence  as  among  the  Greeks. 
Their  acquisitions  in  this  domain  were  in  part  trans- 
planted to  foreign  soil,  yet  not  so  that  (with  the 
possible  exception  of  arithmetic  calculation)  anything 
essentially  new  resulted.  Frequently  what  was  in- 
herited from  the  Greeks  was  not  even  fully  under- 
stood, and  therefore  remained  buried  in  the  literature 
of  the  foreign  nation.  From  the  time  of  the  Renais- 
sance, however,  but  especially  from  that  of  Descartes, 
an  entirely  new  epoch  with  more  powerful  resources 
investigated  the  ancient  treasures  and  laid  them  under 
contribution. 

Among  the  Romans  independent  investigation  of 
mathematical  truths  almost  wholly  disappeared.  What 
they  obtained  from  the  Greeks  was  made  to  serve 
practical  ends  exclusively.  For  this  purpose  parts  of 
Euclid  and  Heron  were  translated.  To  simplify  the 
work  of  the  surveyors  or  agrimensores,  important  geo- 
metric theorems  were  collected  into  a  larger  work  of 
which  fragments  are  preserved  in  the  Codex  Arceri- 


GEOMETRY.  215 

anus.  In  the  work  of  Vitruvius  on  architecture 
(c.  — 14)  is  found  the  value  7r  =  3£  which,  though  less 
accurate  than  Heron's  value  ir  =  3^,  was  more  easily 
employed  in  the  duodecimal  system.*  Boethius  has 
left  a  special  treatise  on  geometry,  but  the  contents 
are  so  paltry  that  it  is  safe  to  assume  that  he  made 
use  of  an  earlier  imperfect  treatment  of  Greek  geom- 
etry. 

Although  the  Hindu  geometry  is  dependent  upon 
the  Greek,  yet  it  has  its  own  peculiarities  due  to  the 
arithmetical  modes  of  thought  of  the  people.  Certain 
parts  of  the  Culvasutras  are  geometric.  These  teach 
the  rope-stretching  already  known  to  the  Egyptians, 
i.  e.,  they  require  the  construction  of  a  right  angle  by 
means  of  a  rope  divided  by  a  knot  into  segments  15 
and  39  respectively,  the  ends  being  fastened  to  a  seg- 
ment 36  (152+  363  =  392).  They  also  use  the  gnomon 
and  deal  with  the  transformation  of  figures  and  the 
application  of  the  Pythagorean  theorem  to  the  multi- 
plication of  a  given  square.  Instead  of  the  quadrature 
of  the  circle  appears  the  circulature  of  the  square,  f 
i.  e.,  the  construction  of  a  circle  equal  to  a  given 
square.  Here  the  diameter  is  put  equal  to  $  of  the 
diagonal  of  the  square,  whence  follows  »r  =  3£  (the 
value  used  among  the  Romans).  In  other  cases  a 
process  is  carried  on  which  yields  the  value  w  =  3. 

The  writings  of  Aryabhatta  contain  certain  incor- 
rect formulae  for  the  mensuration  of  the  pyramid  and 

*Cantor,  I.,  p.  508.  t  Cantor,  I.,  p.  6ot. 


21  6  HISTORY  OF  MATHEMATICS. 


sphere  (for  the  pyramid  V=%Bh},  but  also  a  number 
of  perfectly  accurate  geometric  theorems.  Aryabhatta 
gives  the  approximate  value  ir  =  fff§£  =  3.1416. 
Brahmagupta  teaches  mensurational  or  Heronic  ge- 
ometry and  is  familiar  with  the  formula  for  the  area 
of  the  triangle, 


and  the  formula  for  the  area  of  the  inscribed  quadri- 
lateral, 


which  he  applies  incorrectly  to  any  quadrilateral.  In 
his  work  besides  TT  — 3  we  also  find  the  value  Tr  =  l/10, 
but  without  any  indication  as  to  how  it  was  obtained. 
Bhaskara  likewise  devotes  himself  only  to  algebraic 
geometry.  For  ir  he  gives  not  only  the  Greek  value 
Zf-  and  that  of  Aryabhatta  ff£$f,  but  also  a  value 
ir  =  ||£  =  3.14166  ...  Of  geometric  demonstrations 
Bhaskara  knows  nothing.  He  states  the  theorem, 
adds  the  figure  and  writes  "Behold  !"* 

In  Bhaskara  a  transfer  of  geometry  from  Alexan- 
dria to  India  is  undoubtedly  demonstrable,  and  per- 
haps this  influence  extended  still  further  eastward  to 
the  Chinese.  In  a  Chinese  work  upon  mathematics, 
composed  perhaps  several  centuries  after  Christ,  the 
Pythagorean  theorem  is  applied  to  the  triangle  with 
sides  3,  4,  5 ;  rope-stretching  is  indicated ;  the  ver 

*Cantor,  I.,  p.  614. 


GEOMETRY.  217 

tices  of  a  figure  are  designated  by  letters  after  the 
Greek  fashion ;  IT  is  put  equal  to  3,  and  toward  the 
end  of  the  sixth  century  to  ^-. 

Greek  geometry  reached  the  Arabs  in  part  directly 
and  in  part  through  the  Hindus.  The  esteem,  how- 
ever, in  which  the  classic  works  of  Greek  origin  were 
held  could  not  make  up  for  the  lack  of  real  produc- 
tive power,  and  so  the  Arabs  did  not  succeed  in  a 
single  point  in  carrying  theoretic  geometry,  even  in 
the  subject  of  conic  sections,  beyond  what  had  been 
reached  in  the  golden  age  of  Greek  geometry.  Only 
a  few  particulars  may  be  mentioned.  In  Al  Khowa- 
razmi  is  found  a  proof  of  the  Pythagorean  theorem 
consisting  only  of  the  separation  of  a  square  into 
eight  isosceles  right-angled  triangles.  On  the  whole 
Al  Khowarazmi  draws  more  from  Greek  than  from 
Hindu  sources.  The  classification  of  quadrilaterals 
is  that  of  Euclid ;  the  calculations  are  made  after 
Heron's  fashion.  Besides  the  Greek  value  TT  =  ^-  we 
find  the  Hindu  values  «-  =  .J${$2  and  ir  =  V/10.  Abul 
Wafa  wrote  a  book  upon  geometric  constructions. 
In  this  are  found  combinations  of  several  squares  into 
a  single  one,  as  well  as  the  construction  of  polyhedra 
after  the  methods  of  Pappus.  After  the  Greek  fash- 
ion the  trisection  of  the  angle  occupied  the  attention 
of  Tabit  ibn  Kurra,  Al  Kuhi,  and  Al  Sagani.  Among 
later  mathematicians  the  custom  of  reducing  a  geo- 
metric problem  to  the  solution  of  an  equation  is  com- 
mon. It  was  thus  that  the  Arabs  by  geometric  solu- 


2l8  HISTORY  OF  MATHEMATICS. 

tions  attained  some  excellent  results,  but  results  of  no 
theoretic  importance. 


E.    FOURTH  PERIOD. 

FROM  GERBERT  TO  DESCARTES. 

Among  the  Western  nations  we  find  the  first  traces 
of  geometry  in  the  works  of  Gerbert,  afterward  known 
as  Pope  Sylvester  II.  Gerbert,  as  it  seems,  depends 
upon  the  Codex  Arcerianus,  but  also  mentions  Pyth- 
agoras and  Eratosthenes.*  We  find  scarcely  anything 
here  besides  field  surveying  as  in  Boethius.  Some- 
thing more  worthy  first  appears  in  Leonardo's  (Fibo- 
nacci's) Practica  geometriae\  of  1220,  in  which  work 
reference  is  made  to  Euclid,  Archimedes,  Heron,  and 
Ptolemy.  The  working  over  of  the  material  handed 
down  from  the  ancients,  in  Leonardo's  book,  is  fairly 
independent.  Thus  the  rectification  of  the  circle 
shows  where  this  mathematician,  without  making  use 
of  Archimedes,  determines  from  the  regular  polygon 
of  96  sides  the  value  *=  |||5  =  3.1418. 

Since  among  the  ancients  no  proper  theory  of  star 
polygons  can  be  established,  it  is  not  to  be  wondered 
at  that  the  early  Middle  Ages  have  little  to  show  in 
this  direction.  Star-polygons  had  first  a  mystic  sig 
nificance  only ;  they  were  used  in  the  black  art  as  the 
pentacle,  and  also  in  architecture  and  heraldry.  Adel- 

*  Cantor,  I.,  p.  810.  tHankel,  p.  344. 


GEOMETRY.  2IQ 

ard  of  Bath  went  with  more  detail  into  the  study  of 
star-polygons  in  his  commentary  on  Euclidean  geom- 
etry ;  the  theory  of  these  figures  is  first  begun  by  Re- 
giomontanus. 

The  first  German  mathematical  work  is  the  Deut 
sche  Sphara  written  in  Middle  High  German  by  Conrad 
of  Megenberg,  probably  in  Vienna  in  the  first  half  of 
the  fourteenth  century.  The  first  popular  introduc- 
tion to  geometry  appeared  anonymously  in  the  fif- 
teenth century,  in  six  leaves  of  simple  rules  of  con- 
struction for  geometric  drawing.  The  beginning,  con- 
taining the  construction  of  BC  perpendicular  to  AB 
by  the  aid  of  the  right-angled  triangle  ABC  in  which 
BE  bisects  the  hypotenuse  AC,  runs  as  follows:* 
"From  geometry  some  useful  bits  which  are  written 
after  this.  1.  First  to  make  a  right  angle  quickly. 
Draw  two  lines  across  each  other  just  about  as  you 
wish  and  where  the  lines  cross  each  other  there  put 
an  e.  Then  place  the  compasses  with  one  foot  upon 
the  point  e,  and  open  them  out  as  far  as  you  wish, 
and  make  upon  each  line  a  point.  Let  these  be  the 
letters  a,  b,  c,  all  at  one  distance.  Then  make  a  line 
from  a  to  b  and  from  b  to  c.  So  you  have  a  right  angle 
of  which  here  is  an  example." 

This  construction  of  a  right  angle,  not  given  in 
Euclid  but  first  in  Proclus,  appears  about  the  year 
1500  to  be  in  much  more  extensive  use  than  the 
method  of  Euclid  by  the  aid  of  the  angle  inscribed  in 

*  Gunther,  p.  347. 


220  HISTORY  OF  MATHEMATICS. 

a  semi-circle.  By  his  knowledge  of  this  last  construc- 
tion Adam  Riese  is  said  to  have  humiliated  an  archi- 
tect who  knew  how  to  draw  a  right  angle  only  by  the 
method  of  Proclus. 

Very  old  printed  works  on  geometry  in  German  are  Dz  Puech- 
len  der  fialen  gerechtikait  by  Mathias  Roriczer  (1486)  and  Al- 
brecht  Diirer's  Underiveysung  der  messung  mit  dem  zirckel 
und  richtscheyt  (Nuremberg,  1525).  The  former  gives  in  rather 
unscientific  manner  rules  for  a  special  problem  of  Gothic  architec- 
ture ;  the  latter,  however,  is  a  far  more  original  work  and  on  that 
account  possesses  more  interest.* 

With  the  extension  of  geometric  knowledge  in 
Germany  Widmann  and  Stifel  were  especially  con- 
cerned. Widmann's  geometry,  like  the  elements  of 
Euclid,  begins  with  explanations  :  "  Punctus  is  a  small 
thing  that  cannot  be  divided.  Angulus  is  a  corner 
which  is  made  there  b)'  two  lines,  "f  Quadrilaterals 
have  Arab  names,  a  striking  evidence  that  the  ancient 
Greek  science  was  brought  into  the  West  by  Arab  in 
fluence.  Nevertheless,  by  Roman  writers  (Boethius) 
Widmann  is  led  into  many  errors,  as,  e.  g.,  when  he 
gives  the  area  of  the  isosceles  triangle  of  side  a  as  \a^. 

In  Rudolff's  Coss,  in  the  theory  of  powers,  Stifel 
has  occasion  to  speak  of  a  subject  which  first  receives 
proper  estimation  in  the  modern  geometry,  viz.,  the 
right  to  admit  more  than  three  dimensions.  "Since, 
however,  we  are  in  arithmetic  where  it  is  permitted 
to  invent  many  things  that  otherwise  have  no  form, 

*Gvat\MmSMomilch'sZeitschrJ/t,  XX.,  HI.  2. 

+  Gerhardt.  Geschichte  der  Mathematik  in  Deutschland,  1877. 


GEOMETRY.  221 

this  also  is  permitted  which  geometry  does  not  allow, 
namely  to  assume  solid  lines  and  surfaces  and  go  be- 
yond the  cube  just  as  if  there  were  more  than  three 
dimensions,  which  is,  of  course,  against  nature.  .  .  . 
But  we  have  such  good  indulgence  on  account  of  the 
charming  and  wonderful  usage  of  Coss."* 

Stifel  after  the  manner  of  Ptolemy  extends  the 
study  of  regular  polygons  and  after  the  manner  of 
Euclid  the  construction  of  regular  solids.  He  dis- 
cusses the  quadrature  of  the  circle,  considering  the 
latter  as  a  polygon  of  infinitely  many  sides,  and  de- 
clares the  quadrature  impossible.  According  to  Al- 
brecht  Durer's  Under wey sung,  etc.,  the  quadrature  of 
the  circle  is  obtained  when  the  diagonal  of  the  square 
contains  ten  parts  of  which  the  diameter  of  the  circle 
contains  eight,  i.  e.,  Tr  =  3£.  It  is  expressly  stated, 
however,  that  this  is  only  an  approximate  construc- 
tion. "We  should  need  to  know  quadrature,  circuit, 
that  is  the  making  equal  a  circle  and  a  square,  so  that 
the  one  should  contain  as  much  as  the  other,  but  this 
has  not  yet  been  demonstrated  mechanically  by  schol- 
ars ;  but  that  is  merely  incidental ;  therefore  so  that 
in  practice  it  may  fail  only  slightly,  if  at  all,  they  may 
be  made  equal  as  follows,  f 

*  Stifel,  Die  Coss  Christoffs  Rudolffs.  Mit  schCnen  Exempeln  der  Coss. 
Durch  Michael  Stifel  Gebessert  vnd  sehr  gemehrt.  .  .  .  Gegeben  zum  Haber- 
sten  |  bei  Konigsberg  in  Preussen  |  den  letzten  tag  dess  Herbstmonds  |  im 
Jar  1552.  .  .  .  Zn  Amsterdam  Getruckt  bey  Wilhem  Janson.  Im  Jar  1615. 

tDurer,  Underweysung  der  messung  mit  dent  zirckel  vnd  richttcheyt  in 
Linien  ebnen  vnd  gantzen  corporen.  Durch  Albrecht  Diirer  zusamen  getzogn 
vnd  zu  nutz  alln  kunstlieb  habenden  mit  zu  gehorigen  figuren  in  truck 
gebracht  im  jar  MDXXV.  (Consists  of  vier  Biichlein.) 


222  HISTORY  OF  MATHEMATICS. 

Upon  the  mensuration  of  the  circle*  there  appeared  in  1584  a 
work  by  Simon  van  der  Eycke  in  which  the  value  IT  =  — —  was 

4o4 

given.  By  calculating  the  side  of  the  regular  polygon  of  192  sides 
Ludolph  van  Ceulen  found  (probably  in  1585)  that  7r<3.14205< 
— .  In  his  reply  Simon  v.  d.  Eycke  determined  IT  =  3. 1446055, 
whereupon  L.  v.  Ceulen  in  1586  computed  TT  between  3.142732 
and  3.14103.  Ludolph  van  Ceulen's  papers  contain  a  value  of  TT 
to  35  places,  and  this  value  of  the  Ludolphian  number  was  put 
upon  his  tombstone  (no  longer  known)  in  St.  Peter's  Church  in 
Leyden.  Ceulen's  investigations  led  Snellius,  Huygens,  and  others 
to  further  studies.  By  the  theory  of  rapidly  converging  series  it 
was  first  made  possible  to  compute  TT  to  500  and  more  decimals. f 

A  revival  of  geometry  accompanied  the  activity  of 
Vieta  and  Kepler.  With  these  investigators  begins  a 
period  in  which  the  mathematical  spirit  commences 
to  reach  out  beyond  the  works  of  the  ancients.  J  Vieta 
completes  the  analytic  method  of  Plato;  in  an  ingeni 
ous  way  he  discusses  the  geometric  construction  of 
roots  of  equations  of  the  second  and  third  degrees ; 
he  also  solves  in  an  elementary  manner  the  problem 
of  the  circle  tangent  to  three  given  circles.  Still 
more  important  results  are  secured  by  Kepler.  For 
him  geometry  furnishes  the  key  to  the  secrets  of  the 
world.  With  sure  step  he  follows  the  path  of  indue 
tion  and  in  his  geometric  investigations  freely  con- 
forms to  Euclid.  Kepler  established  the  symbolism 
of  the  "golden  section,"  that  problem  of  Eudoxus 

*Rudio,  F.,  Das  Problem  von  der  Quadratur  des  ZirkeU,  Zurich,  1890 
tD.  Bierens  de  Haan  in  Memo.  Arch.,  I.;  Cantor,  II.,  p.  551. 
tChasles. 


GEOMETRY.  223 

stated  in  the  sixth  book  of  Euclid's  Elements :  "To 
divide  a  limited  straight  line  in  extreme  and  mean 
ratio."*  This  problem,  for  which  Kepler  introduced 
the  designation  sectio  divina  as  well  as  proportio  divina, 
is  in  his  eyes  of  so  great  importance  that  he  expresses 
himself:  "Geometry  has  two  great  treasures:  one  is 
the  theorem  of  Pythagoras,  the  other  the  division  of 
a  line  in  extreme  and  mean  ratio.  The  first  we  may 
compare  to  a  mass  of  gold,  the  second  we  may  call  a 
precious  jewel." 

The  expression  "golden  section"  is  of  more  modern  origin. 
It  occurs  in  none  of  the  text-books  of  the  eighteenth  century  and 
appears  to  have  been  formed  by  a  transfer  from  ordinary  arithme- 
tic. In  the  arithmetic  of  the  sixteenth  and  seventeenth  centuries 
the  rule  of  three  is  frequently  called  the  "golden  rule."  Since  the 
beginning  of  the  nineteenth  century  this  golden  rule  has  given  way 
more  and  more  before  the  so-called  Schlussrechnen  (analysis)  of 
the  Pestalozzi  school.  Consequently  in  place  of  the  "golden  rule," 
which  is  no  longer  known  to  the  arithmetics,  there  appeared  in  the 
elementary  geometries  about  the  middle  of  the  nineteenth  century 
the  "golden  section,"  probably  in  connection  with  contemporary 
endeavors  to  attribute  to  this  geometric  construction  the  impor- 
tance of  a  natural  law. 

Led  on  by  his  astronomical  speculations,  Kepler 
made  a  special  study  of  regular  polygons  and  star- 
polygons.  He  considered  groups  of  regular  polygons 
capable  of  elementary  construction,  viz. ,  the  series  of 
polygons  with  the  number  of  sides  given  by  4*2", 
3-2",  5-2",  15-2"  (from  «  =  0  on),  and  remarked  that 

*  Sonnenburg,  Der  goldene  Schnitt,  1881. 


224  HISTORY  OF  MATHEMATICS. 

a  regular  heptagon  cannot  be  constructed  by  the  help 
of  the  straight  line  and  circle  alone.  Further  there  is 
no  doubt  that  Kepler  well  understood  the  Conies  of 
Apollonius  and  had  experience  in  the  solution  of  prob- 
lems by  the  aid  of  these  curves.  In  his  works  we 
first  find  the  term  "foci"  for  those  points  of  conic 
sections  which  in  earlier  usage  are  known  as  puncta 
ex  comparatione,  puncta  ex  applicatione  facta,  umbilici, 
or  "poles";*  also  the  term  "eccentricity"  for  the 
distance  from  a  focus  to  the  center  divided  by  the 
semi-major  axis,  of  the  curve  of  the  second  order,  and 
the  name  "eccentric  anomaly"  for  the  angle  P'OA, 
where  OA  is  the  semi-major  axis  of  an  ellipse  and  F 
the  point  in  which  the  ordinate  of  a  point  P  on  the 
curve  intersects  the  circle  upon  the  major  axis.f 

Also  in  stereometric  investigations,  which  had  been 
cultivated  to  a  decided  extent  by  Diirer  and  Stifel, 
Kepler  is  preeminent  among  his  contemporaries.  In 
his  Harmonice  Mundi  he  deals  not  simply  with  the 
five  regular  Platonic  and  thirteen  semi-regular  Archi- 
medean solids,  but  also  with  star-polygons  and  star- 
dodecahedra  of  twelve  and  twenty  vertices.  Besides 
this  we  find  the  determination  of  the  volumes  of  solids 
obtained  by  the  revolution  of  conies  about  diameters, 
tangents,  or  secants.  Similar  determinations  of  vol- 
umes were  effected  by  Cavalieri  and  Guldin.  The 
former  employed  a  happy  modification  of  the  method 

*C.  Taylor,  in  Cambr.  Proc.,  IV. 

t  Baltzer,  R.,  Analytische  Geometrie,  1882. 


GEOMETRY.  225 

of  exhaustions,  the  latter  used  a  rule  already  known 
to  Pappus  but  not  accurately  established  by  him. 

To  this  period  belong  the  oldest  known  attempts 
to  solve  geometric  problems  with  only  one  opening  of 
the  compasses,  an  endeavor  which  first  found  accurate 
scientific  expression  in  Steiner's  Geometrische  Con- 
struktionen,  ausgefiihrt  mittels  der  geraden  Linie  und 
eines  festen  Kreises  (1833).  The  first  traces  of  such 
constructions  go  back  to  Abul  Wafa.*  From  the  Arabs 
they  were  transmitted  to  the  Italian  school  where  they 
appear  in  the  works  of  Leonardo  da  Vinci  and  Cardan. 
The  latter  received  his  impulse  from  Tartaglia  who 
used  processes  of  this  sort  in  his  problem-duel  with 
Cardan  and  Ferrari.  They  also  occur  in  the  Resolutio 
omnium  Euclidis  problematum  (Venice,  1553)  of  Bene- 
dictis,  a  pupil  of  Cardan,  in  the  Geometria  deutsch  and 
in  the  construction  of  a  regular  pentagon  by  Durer. 
In  his  Underweysung,  etc.,  Durer  gives  a  geometrically 
accurate  construction  of  the  regular  pentagon  but  also 
an  approximate  construction  of  the  same  figure  to  be 
made  with  a  circle  of  fixed  radius. 

This  method  of  constructing  a  regular  pentagon  on  AB  is  as 
follows :  About  A  and  B  as  centers,  with  radius  AB,  construct  cir- 
cles intersecting  in  C  and  D.  The  circle  about  D  as  a  center  with 
the  same  radius  cuts  the  circles  with  centers  at  A  and  B  in  E  and 
F  and  the  common  chord  CD  in  G.  The  same  circles  are  cut  by 
FG  and  EG  in  /  and  H.  AJ  and  BH  are  sides  of  the  regular 
pentagon.  (The  calculation  of  this  symmetric  pentagon  shows 

*  Gunther  in  Schlomilck's  Zeitschrift,  XX.     Cantor,  I.,  p.  700. 


226 


HISTORY  OF  MATHEMATICS. 


=  W8°20',   while  the   corresponding   angle  of   the   regular 
pentagon  is  103°.) 


In  Dtirer  and  all  his  successors  who  write  upon  rules  of  geo- 
metric construction,  we  find  an  approximate  construction  of  the 
regular  heptagon  :  ' '  The  side  of  the  regular  heptagon  is  half  that 
of  the  equilateral  triangle,"  while  from  calculation  the  half  side 
of  the  equilateral  triangle  =0.998  of  the  side  of  the  heptagon. 
Daniel  Schwenter  likewise  gave  constructions  with  a  single  opening 
of  the  compasses  in  his  Geometria  •practica  nova  et  aitcla  (1625). 
BUrer,  as  is  manifest  from  his  work  Underweysung  der  messung, 
etc.,  already  cited  several  times,  also  rendered  decided  service  in 
the  theory  of  higher  curves.  He  gave  a  general  conception  of  the 
notion  of  asymptotes  and  found  as  new  forms  of  higher  curves  cer- 
tain cyclic  curves  and  mussel-shaped  lines. 

From  the  fifteenth  century  on,  the  methods  of  pro- 
jection make  a  further  advance.  Jan  van  Eyck*  in 
the  great  altar  painting  in  Ghent  makes  use  of  the 
laws  of  perspective,  e.  g.,  in  the  application  of  the 


GEOMETRY.  227 

vanishing  point,  but  without  a  mathematical  grasp  of 
these  laws.  This  is  first  accomplished  by  Albrecht 
Diirer  who  in  his  Underweysung  der  messung  mit  dem 
zirckel  und  richtscheyt  makes  use  of  the  point  of  sight 
and  distance-point  and  shows  how  to  construct  the 
perspective  picture  from  the  ground  plan  and  eleva- 
tion. In  Italy  perspective  was  developed  by  the  archi- 
tect Brunelleschi  and  the  sculptor  Donatello.  The 
first  work  upon  this  new  theory  is  due  to  the  architect 
Leo  Battista  Alberti.  In  this  he  explains  the  perspec- 
tive image  as  the  intersection  of  the  pyramid  of  visual 
rays  with  the  picture-plane.  He  also  mentions  an  in- 
strument for  constructing  it,  which  consists  of  a  frame 
with  a  quadratic  net-work  of  threads  and  a  similar 
net-work  of  lines  upon  the  drawing  surface.  He  also 
gives  the  method  of  the  distance-point  as  invented  by 
him,  by  means  of  which  he  then  pictures  the  ground 
divided  into  quadratic  figures.*  This  process  received 
a  further  extension  at  the  hands  of  Piero  della  Fran- 
cesca  who  employed  the  vanishing  points  of  arbitrary 
horizontal  lines. 

In  German  territory  perspective  was  cultivated 
with  special  zeal  in  Nuremberg  where  the  goldsmith 
Lencker,  some  decades  after  Durer,  extended  the  lat- 
ter's  methods.  The  first  French  study  of  perspective 
is  due  to  the  artist  J.  Cousin  (1560)  who  in  his  Livre 
de  la  perspective  made  use  of  the  point  of  sight  and  the 
distance-point,  besides  the  vanishing  points  of  hori- 


228  HISTORY  OF  MATHEMATICS. 

zontal  lines,  after  the  manner  of  Piero.  Guido  Ubaldi 
goes  noticeably  further  when  he  introduces  the  van- 
ishing point  of  series  of  parallel  lines  of  arbitrary  di- 
rection. What  Ubaldi  simply  foreshadows,  Simon 
Stevin  clearly  grasps  in  its  principal  features,  and  in 
an  important  theorem  he  lays  the  foundation  for  the 
development  of  the  theory  of  collineation. 


F.    FIFTH  PERIOD. 

FROM  DESCARTES  TO  THE  PRESENT. 

Since  the  time  of  Apollonius  many  centuries  had 
elapsed  and  yet  no  one  had  succeeded  in  reaching  the 
full  height  of  Greek  geometry.  This  was  partly  be 
cause  the  sources  of  information  were  relatively  few, 
and  attainable  indirectly  and  with  difficulty,  and  partly 
because  men,  unfamiliar  with  Greek  methods  of  in- 
vestigation, looked  upon  them  with  devout  astonish- 
ment. From  this  condition  of  partial  paralysis,  and 
of  helpless  endeavor  longing  for  relief,  geometry  was 
delivered  by  Descartes.  This  was  not  by  a  simple  ad- 
dition of  related  ideas  to  the  old  geometry,  but  merely 
by  the  union  of  algebra  with  geometry,  thus  giving 
rise  to  analytic  geometry. 

By  way  of  preparation  many  mathematicians,  first 
of  all  Apollonius,  had  referred  the  most  important  ele- 
mentary curves,  namely  the  conies,  to  their  diameters 
and  tangents  and  had  expressed  this  relation  by  equa- 


GEOMETRY.  22Q 

tions  of  the  first  degree  between  areas,  so  that  cer- 
tain relations  were  obtained  between  line-segments 
identical  with  abscissas  and  ordinates. 

In  the  conies  of  Apollonius  we  find  expressions 
which  have  been  translated  "ordinatim  applicatae" 
and  "abscissae."  For  the  former  expression  Fermat 
used  "applicate"  while  others  wrote  "ordinate." 
Since  the  time  of  Leibnitz  (1692)  abscissas  and  ordi- 
nates have  been  called  "co-ordinates."* 

Even  in  the  fourteenth  century  we  find  as  an  ob- 
ject of  study  in  the  universities  a  kind  of  co-ordinate 
geometry,  the  "latitudines  formarum."  "Latitudo"f 
signified  the  ordinate,  "longitude"  the  abscissa  of  a 
variable  point  referred  to  a  system  of  rectangular  co- 
ordinates, and  the  different  positions  of  this  point 
formed  the  "figura."  The  technical  words  longitude 
and  latitude  had  evidently  been  borrowed  from  the 
language  of  astronomy.  In  practice  of  this  art  Oresme 
confined  himself  to  the  first  quadrant  in  which  he 
dealt  with  straight  lines,  circles,  and  even  the  para- 
bola, but  always  so  that  only  a  positive  value  of  a  co- 
ordinate was  considered. 

Among  the  predecessors  of  Descartes  we  reckon, 
besides  Apollonius,  especially,  Vieta,  Oresme,  Cava- 
lieri,  Roberval,  and  Fermat,  the  last  the  most  distin- 
guished in  this  field ;  but  nowhere,  even  by  Fermat, 
had  any  attempt  been  made  to  refer  several  curves  of 

*Baltzer,  R.,  Analytische  Geometric,  1882. 
t  Giinther,  p.  181. 


230  HISTORY  OF  MATHEMATICS. 

different  orders  simultaneously  to  one  system  of  co- 
ordinates, which  at  most  possessed  special  significance 
for  one  of  the  curves.  It  is  exactly  this  thing  which 
Descartes  systematically  accomplished. 

The  thought  with  which  Descartes  made  the  laws 
of  arithmetic  subservient  to  geometry  is  set  forth  by 
himself  in  the  following  manner :  * 

"All  problems  of  geometry  may  be  reduced  to  such 
terms  that  for  their  construction  we  need  only  to  know 
the  length  of  certain  right  lines.  And  just  as  arith- 
metic as  a  whole  comprises  only  four  or  five  opera- 
tions, viz.,  addition,  subtraction,  multiplication,  divi- 
sion, and  evolution,  which  may  be  considered  as  a 
kind  of  division,  so  in  geometry  to  prepare  the  lines 
sought  to  be  known  we  have  only  to  add  other  lines 
to  them  or  subtract  others  from  them ;  or,  having  one 
which  I  call  unity  (so  as  better  to  refer  it  to  numbers), 
which  can  ordinarily  be  taken  at  pleasure,  having  two 
others  to  find  a  fourth  which  shall  be  to  one  of  these 
as  the  other  is  to  unity,  which  is  the  same  as  multi- 
plication ;f  or  to  find  a  fourth  which  shall  be  to  one 
of  the  two  as  unity  is  to  the  other  which  is  the  same 
as  division ; J  or  finally  to  find  one  or  two  or  several 
mean  proportionals  between  unity  and  any  other  line, 
which  is  the  same  as  to  extract  the  square,  cube,  .  .  . 
root.§  I  shall  not  hesitate  to  introduce  these  terms 

*  Marie,  M.,  Histoire  ties  Sciences  Mathematiques  et  Physiques,  1883-1887. 
1'C  :a  =  b:  i,  c  —  ab. 


GEOMETRY.  231 

of  arithmetic  into  geometry  in  order  to  render  myself 
more  intelligible.  It  should  be  observed  that,  by  a2, 
b*,  and  similar  quantities,  I  understand  as  usual  sim- 
ple lines,  and  that  I  call  them  square  or  cube  only  so 
as  to  employ  the  ordinary  terms  of  algebra."  (a2  is 
the  third  proportional  to  unity  and  a,  or  1 :  a  =  a  :  #2, 
and  similarly  b  :  &  =  t>*  :  P. ) 

This  method  of  considering  arithmetical  expres- 
sions was  especially  influenced  by  the  geometric  dis- 
coveries of  Descartes.  As  Apollonius  had  already  de- 
termined points  of  a  conic  section  by  parallel  chords, 
together  with  the  distances  from  a  tangent  belonging 
to  the  same  system,  measured  in  the  direction  of  the 
conjugate  diameter,  so  with  Descartes  every  point  of 
a  curve  is  the  intersection  of  two  straight  lines.  Apol- 
lonius and  all  his  successors,  however,  apply  such 
systems  of  parallel  lines  only  occasionally  and  that  for 
the  sole  purpose  of  presenting  some  definite  property 
of  the  conies  with  especial  distinctness.  Descartes, 
on  the  contrary,  separates  these  systems  of  parallel 
lines  from  the  curves,  assigns  them  an  independent 
existence  and  so  obtains  for  every  point  on  the  curve 
a  relation  between  two  segments  of  given  direction, 
which  is  nothing  else  than  an  equation.  The  geo- 
metric study  of  the  properties  of  this  curve  can  then 
be  replaced  by  the  discussion  of  the  equation  after  the 
methods  of  algebra.  The  fundamental  elements  for 
the  determination  of  a  point  of  a  curve  are  its  co-or- 
dinates, and  from  long  known  theorems  it  was  evident 


232  HISTORY  OF  MATHEMATICS. 

that  a  point  of  the  plane  can  be  fixed  by  two  co-ordi- 
nates, a  point  of  space  by  three. 

Descartes's  Geometry  is  not,  perhaps,  a  treatise 
on  analytic  geometry,  but  only  a  brief  sketch  which 
sets  forth  the  foundations  of  this  theory  in  outline. 
Of  the  three  books  which  constitute  the  whole  work 
only  the  first  two  deal  with  geometry ;  the  third  is  of 
algebraic  nature  and  contains  the  celebrated  rule  of 
signs  illustrated  by  a  simple  example,  as  well  as  the 
solution  of  equations  of  the  third  and  fourth  degrees 
with  the  construction  of  their  roots  by  the  use  of 
conies. 

The  first  impulse  to  his  geometric  reflections  was 
due,  as  Descartes  himself  says,  to  a  problem  which 
according  to  Pappus  had  already  occupied  the  atten- 
tion of  Euclid  and  Apollonius.  It  is  the  problem  to 
find  a  certain  locus  related  to  three,  four,  or  several 
lines.  Denoting  the  distances,  measured  in  given  di- 
rections, of  a  point  P  from  the  straight  lines  g\,  gi .  .  . 
gH  by  ei,  <?a .  .  .  en,  respectively,  we  shall  have 

for  three  straight  lines  : 


aes 
for  four  straight  lines  :  — l    2    =  k, 


for  five  straight  lines :    l        *  =  k, 
ae4e6 

and  so  on.  The  Greeks  originated  the  solution  of  the 
first  two  cases,  which  furnish  conic  sections.  No  ex- 
ample could  have  shown  better  the  advantage  of  the 


GEOMETRY.  233 

new  method.  For  the  case  of  three  lines  Descartes 
denotes  a  distance  by  y,  the  segment  of  the  corres- 
ponding line  between  the  foot  of  this  perpendicular 
and  a  fixed  point  by  x,  and  shows  that  every  other 
segment  involved  in  the  problem  can  be  easily  con- 
structed. Further  he  states  "that  if  we  allow  y  to 
grow  gradually  by  infinitesimal  increments,  x  will 
grow  in  the  same  way  and  thus  we  may  get  infinitely 
many  points  of  the  locus  in  question." 

The  curves  with  which  Descartes  makes  us  gradu- 
ally familiar  he  classifies  so  that  lines  of  the  first  and 
second  orders  form  a  first  group,  those  of  the  third 
and  fourth  orders  a  second,  those  of  the  fifth  and 
sixth  orders  a  third,  and  so  on.  Newton  was  the  first 
to  call  a  curve,  which  is  defined  by  an  algebraic  equa- 
tion of  the  «th  degree  between  parallel  co-ordinates,  a 
line  of  the  «th  order,  or  a  curve  of  the  (n — l)th  class. 
The  division  into  algebraic  and  transcendental  curves 
was  introduced  by  Leibnitz ;  previously,  after  the 
Greek  fashion,  the  former  had  been  called  geometric, 
the  latter  mechanical  lines.* 

Among  the  applications  which  Descartes  makes, 
the  problem  of  tangents  is  prominent.  This  he  treats 
in  a  peculiar  way :  Having  drawn  a  normal  to  a  curve 
at  the  point  P,  he  describes  a  circle  through  P  with 
the  center  at  the  intersection  of  this  normal  with  the 

*Baltzer,  R.,  Analytitche  Geometrie,  1882.  Up  to  the  time  of  Descartes 
all  lines  except  straight  lines  and  conies  were  called  mechanical.  He  was 
the  first  to  apply  the  term  geometric  lines  to  curves  of  degree  higher  than 
the  second. 


234  HISTORY  OF  MATHEMATICS. 

A"- axis,  and  asserts  that  this  circle  cuts  the  curve  at  P 
in  two  consecutive  points;  i.  e.,  he  states  the  condi- 
tion that  after  the  elimination  of  x  the  equation  in  y 
shall  have  a  double  root. 

A  natural  consequence  of  the  acceptance  of  the 
Cartesian  co-ordinate  system  was  the  admission  of 
negative  roots  of  algebraic  equations.  These  negative 
roots  had  now  a  real  significance ;  they  could  be  rep- 
resented, and  hence  were  entitled  to  the  same  rights 
as  positive  roots. 

In  the  period  immediately  following  Descartes, 
geometry  was  enriched  by  the  labors  of  Cavalieri, 
Fermat,  Roberval,  Wallis,  Pascal,  and  Newton,  not 
at  first  by  a  simple  application  of  the  co-ordinate  ge- 
ometry, but  often  after  the  manner  of  the  ancient 
Greek  geometry,  though  with  some  of  the  methods 
essentially  improved.  The  latter  is  especially  true  of 
Cavalieri,  the  inventor  of  the  method  of  indivisibles,* 
which  a  little  later  was  displaced  by  the  integral  ca'- 
culus,  but  may  find  a  place  here  since  it  rendered  ser- 
vice to  geometry  exclusively.  Cavalieri  enjoyed  work- 
ing with  the  geometry  of  the  ancients.  For  example, 
he  was  the  first  to  give  a  satisfactory  proof  of  the  so- 
called  Guldin's  rule  already  stated  by  Pappus.  His 
chief  endeavor  was  to  find  a  general  process  for  the 
determination  of  areas  and  volumes  as  well  as  centers 
of  gravity,'  and  for  this  purpose  he  remodelled  the 

*In  French  works  Mtthode  des  indivisibles,  originally  in  the  work  Geo- 
metria  indivisibilibus  continuorum  nova  quadam  rations  fromota,  Bologna, 
1635. 


GEOMETRY.  235 

method  of  exhaustions.  Inasmuch  as  Cavalieri's 
method,  of  which  he  was  master  as  early  as  1629,  may 
even  to-day  replace  to  advantage  ordinary  integration 
in  elementary  cases,  its  essential  character  may  be  set 
forth  in  brief  outline.* 

If  y=f(x}  is  the  equation  of  a  curve  in  rectangu- 
lar co-ordinates,  and  he  wishes  to  determine  the  area 
bounded  by  the  axis  of  x,  a  portion  of  the  curve,  and 
the  ordinates  corresponding  to  XQ  and  x\,  Cavalieri 
divides  the  difference  x\  —  XQ  into  n  equal  parts.  Let 
h  represent  such  a  part  and  let  n  be  taken  very  large. 
An  element  of  the  surface  is  then  =hy  =  hf(x},  and 
the  whole  surface  becomes 


For  «  =  oo  we  evidently  get  exactly 


But  this  is  not  the  quantity  which  Cavalieri  seeks  to 
determine.  He  forms  only  the  ratios  of  portions  of 
the  area  sought,  to  the  rectangle  with  base  xi  —  x0 
and  altitude  yi,  so  that  the  quantity  to  be  determined 
is  the  following  : 


Cavalieri  applies  this  formula,  which  he  derives  in 


236  HISTORY   OF  MATHEMATICS. 

complete  generality  from  grounds  of  analogy,  only  to 
the  case  where /(jf)  is  of  the  form  Ax?"  (m  =  2,  3,  4). 
The  extension  to  further  cases  was  made  by  Rober 
val,  Wallis,  and  Pascal. 

In  the  simplest  cases  the  method  of  indivisibles  gives  the  fol- 
lowing results.*  For  a  parallelogram  the  indivisible  quantity  or 
element  of  surface  is  a  parallel  to  the  base ;  the  number  of  indi- 
visible quantities  is  proportional  to  the  altitude ;  hence  we  have 
as  the  measure  of  the  area  of  the  parallelogram  the  product  of  the 
measures  of  the  base  and  altitude.  The  corresponding  conclusion 
holds  for  the  prism.  In  order  to  compare  the  area  of  a  triangle 
with  that  of  the  parallelogram  of  the  same  base  and  altitude,  we 
decompose  each  into  elements  by  equidistant  parallels  to  the  base. 
The  elements  of  the  triangle  are  then,  beginning  with  the  least,  1, 
2,  3,  .  .  .  n  ;  those  of  the  parallelogram,  «,»,...».  Hence  the 
ratio 

Triangle       _l-f2  +  ...-fw  =  jn(n-|-l)_l/l\ 
Parallelogram  n~n  «2  t  \         » / ' 

whence  for  n  =  oo  we  get  the  value  $.  For  the  corresponding  solids 
we  get  likewise 

Pyramid  _  l»-f22  +  ..  . +n2  _  %n  (n  +  1)  (2n  -f 1) 
Prism    ~  «8  «» 


After  the  lapse  of  a  few  decades  this  analytic- 
geometric  method  of  Cavalieri's  was  forced  into  the 
background  by  the  integral  calculus,  which  could  be 
directly  applied  in  all  cases.  At  first,  however,  Rober- 
val,  known  by  his  method  of  tangents,  trod  in  the 
footsteps  of  Cavalieri.  Wallis  used  the  works  of  Des- 

*  Marie. 


GEOMETRY.  237 

cartes  and  CaValieri  simultaneously,  and  considered 
especially  curves  whose  equations  were  of  the  form 
y  =  ^",  m  integral  or  fractional,  positive  or  negative. 
His  chief  service  consists  in  this,  that  in  his  brilliant 
work  he  put  a  proper  estimate  upon  Descartes's  dis- 
covery and  rendered  it  more  accessible.  In  this  work 
Wallis  also  defines  the  conies  as  curves  of  the  second 
degree,  a  thing  never  before  done  in  this  definite 
manner. 

Pascal  proved  to  be  a  talented  disciple  of  Cavalieri 
and  Desargues.  In  his  work  on  conies,  composed 
about  1639  but  now  lost  (save  for  a  fragment),*  we 
find  Pascal's  theorem  of  the  inscribed  hexagon  or 
Hexagramma  mysticum  as  he  termed  it,  which  Bessel 
rediscovered  in  1820  without  being  aware  of  Pascal's 
earlier  work, f  also  the  theorem  due  to  Desargues  that 
if  a  straight  line  cuts  a  conic  in  P  and  Q,  and  the 
sides  of  an  inscribed  quadrilateral  in  A,  B,  C,  D,  we 
have  the  following  equation  : 

PA-PC  _  QA-QC 
PB-PD  ~  QC-  QD' 

Pascal's  last  work  deals  with  a  curve  called  by  him 
the  roulette,  by  Roberval  the  trochoid,  and  generally 
known  later  as  the  cycloid.  Bouvelles  (1503)  already 
knew  the  construction  of  this  curve,  as  did  Cardinal 
von  Cusa  in  the  preceding  century.  J  Galileo,  as  is 
shown  by  a  letter  to  Torricelli  in  1639,  had  made  (be 

*  Cantor,  II.,  p.  622.  t  Bianco  in  Torino  Att.,  XXI. 

t  Cantor,  II.,  pp.  186,  351. 


238  HISTORY  OF  MATHEMATICS. 

ginning  in  1590)  an  exhaustive  study  of  rolling  curves 
in  connection  with  the  construction  of  bridge  arches. 
The  quadrature  of  the  cycloid  and  the  determination 
of  the  volume  obtained  by  revolution  about  its  axis 
had  been  effected  by  Roberval,  and  the  construction 
of  the  tangent  by  Descartes.  In  the  year  1658  Pascal 
was  able  to  determine  the  length  of  an  arc  of  a  cy- 
cloidal  segment,  the  center  of  gravity  of  this  surface, 
and  the  corresponding  solid  of  revolution.  Later  the 
cycloid  appears  in  physics  as  the  brachistochrone  and 
tautochrone,  since  it  permits  a  body  sliding  upon  it  to 
pass  from  one  fixed  point  to  another  in  the  shortest 
time,  while  it  brings  a  material  point  oscillating  upon 
it  to  its  lowest  position  always  in  the  same  time. 
Jacob  and  John  Bernoulli,  among  others,  gave  atten- 
tion to  isoperimetric  problems ;  but  only  the  former 
secured  any  results  of  value,  by  furnishing  a  rigid 
method  for  their  solution  which  received  merely  an 
unimportant  simplification  from  John  Bernoulli.  (See 
pages  178-179.) 

The  decades   following  Pascal's  activity  were  in 
large  part  devoted  to  the  study  of  tangent  problems 
and  the  allied  normal  problems,  but  at  the  same  time 
the  general  theory  of   plane   curves  was  constantly 
developing.    Barrow  gave  a  new  method  of  determin 
ing  tangents,  and  Huygens  studied  evolutes  of  curves 
and  indicated  the  way  of  determining  radii  of  curva 
ture.     From  the  consideration  of  caustics,   Tschirn 
hausen  was  led  to  involutes  and  Maclaurin  constructed 


GEOMETRY.  239 

the  circle  of  curvature  at  any  point  of  an  algebraic 
curve.  The  most  important  extension  of  this  theory 
was  made  in  Newton's  Enumeratio  linearum  tertii  or- 
dinis  (1706).  This  treatise  establishes  the  distinction 
between  algebraic  and  transcendental  curves.  It  then 
makes  an  exhaustive  study  of  the  equation  of  a  curve 
of  the  third  order,  and  thus  finds  numerous  such  curves 
which  may  be  represented  as  "shadows  "  of  five  types, 
a  result  which  involves  an  analytic  theory  of  perspec- 
tive. Newton  knew  how  to  construct  conies  from  five 
tangents.  He  came  upon  this  discovery  in  his  en- 
deavor to  investigate  "after  the  manner  of  the  an- 
cients" without  analytic  geometry.  Further  he  con- 
sidered multiple  points  of  a  curve  at  a  finite  distance 
and  at  infinity,  and  gave  rules  for  investigating  the 
course  of  a  curve  in  the  neighborhood  of  one  of  its 
points  ("Newton's  parallelogram"  or  "analytic  tri- 
angle "),  as  also  for  the  determination  of  the  order  of 
contact  of  two  curves  at  one  of  their  common  points. 
(Leibnitz  and  Jacob  Bernoulli  had  also  written  upon 
osculations ;  Plucker  (1831)  called  the  situation  where 
two  curves  have  k  consecutive  points  in  common  "a 
£-pointic  contact";  in  the  same  case  Lagrange  (1779) 
had  spoken  of  a  "contact  of  (k — l)th  order. ")f 

Additional  work  was  done  by  Newton's  disciples, 
Cotes  and  Maclaurin,  as  well  as  by  Waring.  Mac- 
laurin  made  interesting  investigations  upon  corre- 

*  Baltzer. 

tCayley,  A.,  Address  to  the  British  Association,  etc.,  1883. 


840  HISTORY  OF  MATHEMATICS. 

spending  points  of  a  curve  of  the  third  order,  and 
thus  showed  that  the  theory  of  these  curves  was  much 
more  comprehensive  than  that  of  conies.  Euler  like- 
wise entered  upon  these  investigations  in  his  paper 
Sur  une  contradiction  apparente  dans  la  thtorie  des  courbes 
planes  (Berlin,  1748),  where  it  is  shown  that  by  eight  in- 
tersections of  two  curves  of  the  third  order  the  ninth  is 
completely  determined.  This  theorem,  which  includes 
Pascal's  theorem  for  conies,  introduced  point  groups, 
or  systems  of  points  of  intersection  of  two  curves,  into 
geometry.  This  theorem  of  Euler's  was  noticed  in 
1750  by  Cramer  who  gave  special  attention  to  the  sin- 
gularities of  curves  in  his  works  upon  the  intersection 
of  two  algebraic  curves  of  higher  order;  hence  the 
obvious  contradiction  between  the  number  of  points 
determining  a  plane  curve  and  the  number  of  inde- 
pendent intersections  of  two  curves  of  the  same  order 
bears  the  name  of  "Cramer's  paradox."  This  contra- 
diction was  solved  by  Lame1  in  1818  by  the  principle 
which  bears  his  name.*  Partly  in  connection  with 
known  results  of  the  Greek  geometry,  and  partly  in- 
dependently, the  properties  of  certain  algebraic  and 
transcendental  curves  were  investigated.  A  curve 
which  is  formed  like  the  conchoid  of  Nicomedes,  if 
we  replace  the  straight  line  by  a  circle,  is  called  by 


*Loria,  G.,  Die  hauptsachlichsten  Theorien  der  Geometrie  in  ihrer  frvhe 
ren  undjetxigen  Entivicklung.  Deutsch  von  Schiitte,  1888.  For  a  more  accu- 
rate account  of  Cramer's  paradox,  in  which  proper  credit  is  given  to  Mac- 
laurin's  discovery,  see  Scott,  C.  A.,  "  On  the  Intersections  of  Plane  Curves," 
Bull.  Am.  Math.  Soc.,  March,  1898. 


GEOMETRY.  24! 

Roberval  the  limason  of  Pascal.  The  cardioid  of  the 
eighteenth  century  is  a  special  case  of  this  spiral.  If, 
with  reference  to  two  fixed  points  A,  J3,  a  point  P 
satisfies  the  condition  that  a  linear  function  of  the 
distances  PA,  PB  has  a  constant  value,  then  is  the 
locus  of  P  a  Cartesian  oval.  This  curve  was  found  by 
Descartes  in  his  studies  in  dioptrics.  For  PA- PB  = 
constant,  we  have  Cassini's  oval,  which  the  astronomer 
of  Louis  XIV.  wished  to  regard  as  the  orbit  of  a  planet 
instead  of  Kepler's  ellipse.  In  special  cases  Cassini's 
oval  contains  a  loop,  and  this  form  received  from 
Jacob  Bernoulli  (1694)  the  name  lemniscate.  With 
the  investigation  of  the  logarithmic  curve  y  =  a*  was 
connected  the  study  made  by  Jacob  and  John  Ber- 
noulli, Leibnitz,  Huygens,  and  others,  of  the  curve  of 
equilibrium  of  an  inextensible,  flexible  thread.  This 
furnished  the  catenary  (catenaria,  1691),  the  idea  of 
which  had  already  occurred  to  Galileo.*  The  group 
of  spirals  found  by  Archimedes  was  enlarged  in  the 
seventeenth  and  eighteenth  centuries  by  the  addition 
of  the  hyperbolic,  parabolic,  and  logarithmic  spirals, 
and  Cotes's  lituus  (1722).  In  1687  Tschirnhausen  de- 
fined a  quadratrix,  differing  from  that  of  the  Greeks, 
as  the  locus  of  a  point  P,  lying  at  the  same  time  upon 
LQ\\BO  and  upon  MP\\OA  (OAB  is  a  quadrant), 
where  L  moves  over  the  quadrant  and  M  over  the 
radius  OB  uniformly.  Whole  systems  of  curves  and 
surfaces  were  considered.  Here  belong  the  investiga- 

*  Cantor,  III.,  p.  an. 


242  HISTORY  OF  MATHEMATICS. 

tions  of  involutes  and  evolutes,  envelopes  in  general, 
due  to  Huygens,  Tschirnhausen,  John  Bernoulli, 
Leibnitz,  and  others.  The  consideration  of  the  pen- 
cil of  rays  through  a  point  in  the  plane,  and  of  the 
pencil  of  planes  through  a  straight  line  in  space,  was 
introduced  by  Desargues,  1639.* 

The  extension  of  the  Cartesian  co-ordinate  method 
to  space  of  three  dimensions  was  effected  by  the  labors 
of  Van  Schooten,  Parent,  and  Clairaut.f  Parent  rep- 
resented a  surface  by  an  equation  involving  the  three 
co-ordinates  of  a  point  in  space,  and  Clairaut  per- 
fected this  new  procedure  in  a  most  essential  manner 
by  a  classic  work  upon  curves  of  double  curvature. 
Scarcely  thirty  years  later  Euler  established  the  ana- 
lytic theory  of  the  curvature  of  surfaces,  and  the  clas- 
sification of  surfaces  in  accordance  with  theorems 
analogous  to  those  used  in  plane  geometry.  He  gives 
formulae  of  transformation  of  space  co-ordinates  and 
a  discussion  of  the  general  equation  of  surfaces  of  the 
second  order,  with  their  classification.  Instead  of 
Euler's  names  :  "elliptoid,  elliptic-hyperbolic,  hyper- 
bolic-hyperbolic, elliptic- parabolic,  parabolic- hyper- 
bolic surface,"  the  terms  now  in  use,  " ellipsoid,  hyper- 
boloid,  paraboloid,"  were  naturalized  by  Biot  and 
Lacroix.| 

Certain  special  investigations  are  worthy  of  men- 
tion. In  1663  Wallis  studied  plane  sections  and 
effected  the  cubature,  of  a  conoid  with  horizontal  di- 


GEOMETRY.  243 

recting  plane  whose  generatrix  intersects  a  vertical 
directing  straight  line  and  vertical  directing  circle 
(cono-cuneus}.  To  Wren  we  owe  an  investigation  of 
the  hyperboloid  of  revolution  of  two  sheets  (1669) 
which  he  called  "cylindroid."  The  domain  of  gauche 
curves,  of  which  the  Greeks  knew  the  common  helix 
of  Archytas  and  the  spherical  spiral  corresponding  in 
formation  to  the  plane  spiral  of  Archimedes,  found  an 
extension  in  the  line  which  cuts  under  a  constant  an- 
gle the  meridians  of  a  sphere.  Nunez  (1546)  had 
recognized  this  curve  as  not  plane,  and  Snellius  (1624) 
had  given  it  the  name  loxodromia  sphaerica.  The  prob- 
lem of  the  shortest  line  between  two  points  of  a  sur- 
face, leading  to  gauche  curves  which  the  nineteenth 
century  has  termed  "geodetic  lines,"  was  stated  by 
John  Bernoulli  (1698)  and  taken  in  hand  by  him  with 
good  results.  In  a  work  of  Pitot  in  1724  (printed  in 
1726)*  upon  the  helix,  we  find  for  the  first  time  the 
expression  ligne  d  double  courbure,  line  of  double  curva- 
ture, for  a  gauche  curve.  In  1776  and  1780  Meusnier 
gave  theorems  upon  the  tangent  planes  to  ruled  sur- 
faces, and  upon  the  curvature  of  a  surface  at  one  of 
its  points,  as  a  preparation  for  the  powerful  develop- 
ment of  the  theory  of  surfaces  soon  to  begin,  f 

There  are  still  some  minor  investigations  belong- 
ing to  this  period  deserving  of  mention.  The  alge- 
braic expression  for  the  distance  between  the  centers 
of  the  inscribed  and  circumscribed  circles  of  a  triangle 

"Cantor,  III.,  p.  .428.  t  Baltzer. 


244  HISTORY  OF  MATHEMATICS. 

was  determined  by  William  Chappie  (about  1746), 
afterwards  by  Landen  (1755)  and  Euler  (1765).*  In 
1769  Meister  calculated  the  areas  of  polygons  whose 
sides,  limited  by  every  two  consecutive  vertices,  inter- 
sect so  that  the  perimeter  contains  a  certain  number 
of  double  points  and  the  polygon  breaks  up  into  cells 
with  simple  or  multiple  positive  or  negative  areas. 
Upon  the  areas  of  such  singular  polygons  Mobius  pub- 
lished later  investigations  (1827  and  1865).*  Saurin 
considered  the  tangents  of  a  curve  at  multiple  points 
and  Ceva  starting  from  static  theorems  studied  the 
transversals  of  geometric  figures.  Stewart  still  further 
extended  the  theorems  of  Ceva,  while  Cotes  deter- 
mined the  harmonic  mean  between  the  segments  of  a 
secant  to  a  curve  of  the  nth  order  reckoned  from  a 
fixed  point.  Carnot  also  extended  the  theory  of  trans- 
versals. Lhuilier  solved  the  problem  :  In  a  circle  to 
inscribe  a  polygon  of  n  sides  passing  through  n  fixed 
points.  Brianchon  gave  the  theorem  concerning  the 
hexagon  circumscribed  about  a  conic  dualistically  re- 
lated to  Pascal's  theorem  upon  the  inscribed  hexagon. 
The  application  of  these  two  theorems  to  the  surface 
of  the  sphere  was  effected  by  Hesse  and  Thieme.  In 
the  work  of  Hesse  a  Pascal  hexagon  is  formed  upon 
the  sphere  by  six  points  which  lie  upon  the  intersec- 
tion of  the  sphere  with  a  cone  of  the  second  order 
having  its  vertex  at  the  center  of  the  sphere.  Thieme 
selects  a  right  circular  cone.  The  material  usually 

*Fortschritte,  1887,  p.  32.  tBaltzer. 


GEOMETRY.  245 

taken  for  the  elementary  geometry  of  the  schools  has 
among  other  things  received  an  extension  through 
numerous  theorems  upon  the  circle  named  after  K. 
W.  Feuerbach  (1822),  upon  symmedian  lines  of  a 
triangle,  upon  the  Grebe  point  and  the  Brocard  fig- 
ures (discovered  in  part  by  Crelle,  1816;  again  intro- 
duced by  Brocard,  1875).* 

The  theory  of  regular  geometric  figures  received 
its  most  important  extension  at  the  hands  of  Gauss, 
who  discovered  noteworthy  theorems  upon  the  possi- 
bility or  impossibility  of  elementary  constructions  of 
regular  polygons.  (See  p.  160.)  Poinsot  elaborated 
the  theory  of  the  regular  polyhedra  by  publishing  his 
views  on  the  five  Platonic  bodies  and  especially  upon 
the  "  Kepler-Poinsot  regular  solids  of  higher  class," 
viz.,  the  four  star-polyhedra  which  are  formed  from 
the  icosahedron  and  dodecahedron.  These  studies 
were  continued  by  Wiener,  Hessel,  and  Hess,  with 
the  removal  of  certain  restrictions,  so  that  a  whole 
series  of  solids,  which  in  an  extended  sense  may  be 
regarded  as  regular,  may  be  added  to  those  named 
above.  Corresponding  studies  for  four-dimensional 
space  have  been  undertaken  by  Scheffler,  Rudel, 
Stringham,  Hoppe,  and  Schlegel.  They  have  deter- 
mined that  in  such  a  space  there  exist  six  regular  fig- 
ures of  which  the  simplest  has  as  its  boundary  five 
tetrahedra.  The  boundaries  of  the  remaining  five  fig- 

*Lieber,  Ueber  die  Gegenmittellinie,den  Grebe'schen  Punkt  und  den  Bro- 
card'schen  Kreis,  1886-1888. 


246  HISTORY  OF  MATHEMATICS. 

ures  require  16  or  600  tetrahedra,  8  hexahedra,  24  oc- 
tahedra,  120  dodecahedra.*  It  may  be  mentioned 
further  that  in  1849  the  prismatoid  was  introduced 
into  stereometry  by  E.  F.  August,  and  that  Schubert 
and  Stoll  so  generalised  the  Apollonian  contact  prob- 
lem as  to  be  able  to  give  the  construction  of  the  six- 
teen spheres  tangent  to  four  given  spheres. 

Projective  geometry,  called  less  precisely  modern 
geometry  or  geometry  of  position,  is  essentially  a 
creation  of  the  nineteenth  century.  The  analytic  ge- 
ometry of  Descartes,  in  connection  with  the  higher 
analysis  created  by  Leibnitz  and  Newton,  had  regis- 
tered a  series  of  important  discoveries  in  the  domain 
of  the  geometry  of  space,  but  it  had  not  succeeded  in 
obtaining  a  satisfactory  proof  for  theorems  of  pure 
geometry.  Relations  of  a  specific  geometric  character 
had,  however,  been  discovered  in  constructive  draw- 
ing. Newton's  establishment  of  his  five  principal 
types  of  curves  of  the  third  order,  of  which  the  sixty- 
four  remaining  types  may  be  regarded  as  projections, 
had  also  given  an  impulse  in  the  same  direction.  Still 
more  important  were  the  preliminary  works  of  Carnot, 
which  paved  the  way  for  the  development  of  the  new 
theory  by  Poncelet,  Chasles,  Steiner,  and  von  Staudt. 
They  it  was  who  discovered  "the  overflowing  spring 
of  deep  and  elegant  theorems  which  with  astonishing 
facility  united  into  an  organic  whole,  into  the  graceful 
edifice  of  projective  geometry,  which,  especially  with 

*Serret,  Essai  d'une  nouvelle  mfthode,  etc.,  1873. 


GEOMETRY.  247 

reference  to  the  theory  of  curves  of  the  second  order, 
may  be  regarded  as  the  ideal  of  a  scientific  organism."* 

Projective  geometry  found  its  earliest  unfolding  on 
French  soil  in  the  Geometric  descriptive  of  Monge  whose 
astonishing  power  of  imagination,  supported  by  the 
methods  of  descriptive  geometry,  discovered  a  host  of 
properties  of  surfaces  and  curves  applicable  to  the 
classification  of  figures  in  space.  His  work  created 
"for  geometry  the  hitherto  unknown  idea  of  geomet- 
ric generality  and  geometric  elegance,  "f  and  the  im- 
portance of  his  works  is  fundamental  not  only  for  the 
theory  of  projectivity  but  also  for  the  theory  of  the 
curvature  of  surfaces.  To  the  introduction  of  the 
imaginary  into  the  considerations  of  pure  geometry 
Monge  likewise  gave  the  first  impulse,  while  his  pupil 
Gaultier  extended  these  investigations  by  defining  the 
radical  axis  of  two  circles  as  a  secant  of  the  same 
passing  through  their  intersections,  whether  real  or 
imaginary. 

The  results  of  Monge's  school  thus  derived,  which 
were  more  closely  related  to  pure  geometry  than  to 
the  analytic  geometry  of  Descartes,  consisted  chiefly 
in  a  series  of  new  and  interesting  theorems  upon  sur- 
faces of  the  second  order,  and  thus  belonged  to  the 
same  field  that  had  been  entered  upon  before  Monge's 
time  by  Wren  (1669),  Parent  and  Euler.  That  Monge 


*  Brill,  A..  Antrittsrede  in  Tubingen,  1884. 

t  Hankel,  Die  Elemente  der  projektivischen  Geometrie  in  synthetischer  Be- 
kandlung,  1875. 


248  HISTORY  OF  MATHEMATICS. 

did  not  hold  analytic  methods  in  light  esteem  is  shown 
by  his  Application  de  I'algebre  a  la  ge'ome'trie  (1805)  in 
which,  as  Plucker  says,  "he  introduced  the  equation 
of  the  straight  line  into  analytic  geometry,  thus  laying 
the  foundation  for  the  banishment  of  all  constructions 
from  it,  and  gave  it  that  new  form  which  rendered 
further  extension  possible." 

While  Monge  was  working  by  preference  in  the 
space  of  three  dimensions,  Carnot  was  making  a  spe- 
cial study  of  ratios  of  magnitudes  in  figures  cut  by 
transversals,  and  thus,  by  the  introduction  of  the  nega- 
tive, was  laying  the  foundation  for  a  ge'ome'trie  de  posi- 
tion which,  however,  is  not  identical  with  the  Geometric 
der  Lage  of  to-day.  Not  the  most  important,  but  the 
most  noteworthy  contribution  for  elementary  school 
geometry  is  that  of  Carnot's  upon  the  complete  quadri- 
lateral and  quadrangle. 

Monge  and  Carnot  having  removed  the  obstacles 
which  stood  in  the  way  of  a  natural  development  of 
geometry  upon  its  own  territory,  these  new  ideas  could 
now  be  certain  of  a  rapid  development  in  well-pre- 
pared soil.  Poncelet  furnished  the  seed.  His  work, 
Traiti  des  proprie'te's  projectives  des  figures,  which  ap- 
peared in  1822,  investigates  those  properties  of  figures 
which  remain  unchanged  in  projection,  i.  e.,  their  in- 
variant properties.  The  projection  is  not  made  here, 
as  in  Monge,  by  parallel  rays  in  a  given  direction,  but 
by  central  projection,  and  so  after  the  manner  of  per- 
spective. In  this  way  Poncelet  came  to  introduce 


GEOMETRY.  249 

the  axis  of  perspective  and  center  of  perspective  (ac- 
cording to  Chasles,  axis  and  center  of  homology)  in 
the  consideration  of  plane  figures  for  which  Desargues 
had  already  established  the  fundamental  theorems. 
In  1811  Servois  had  used  the  expression  "pole  of  a 
straight  line,"  and  in  1813  Gergonne  the  terms  "polar 
of  a  point"  and  "duality,"  but  in  1818  Poncelet  de- 
veloped some  observations  made  by  Lahire  in  1685. 
upon  the  mutual  correspondence  of  pole  and  polar  in 
the  case  of  conies,  into  a  method  of  transforming  fig- 
ures into  their  reciprocal  polars.  Gergonne  recog- 
nized in  this  theory  of  reciprocal  polars  a  principle 
whose  beginnings  were  known  to  Vieta,  Lansberg, 
and  Snellius,  from  spherical  geometry.  He  called  it 
the  "principle  of  duality"  (1826).  In  1827  Gergonne 
associated  dualistically  with  the  notion  of  order  of  a 
plane  curve  that  of  its  class.  The  line  is  of  the  «th 
order  when  a  straight  line  of  the  plane  cuts  it  in  « 
points,  of  the  nth  class  when  from  a  point  in  the  plane 
11  tangents  can  be  drawn  to  it.* 

While  in  France  Chasles  alone  interested  himself 
thoroughly  in  its  advancement,  this  new  theory  found 
its  richest  development  in  the  third  decade  of  the 
nineteenth  century  upon  German  soil,  where  almost 
at  the  same  time  the  three  great  investigators,  Mobius, 
Plucker,  and  Steiner  entered  the  field.  From  this 
time  on  the  synthetic  and  more  constructive  tendency 
followed  by  Steiner,  von  Staudt,  and  Mobius  diverges! 

*Baltzer.  t  Brill,  A.,  AntrittsreeU  in  Tubingen,  1884. 


250  HISTORY  OF  MATHEMATICS. 

from  the  analytic  side  of  the  modern  geometry  which 
Plucker,  Hesse,  Aronhold,  and  Clebsch  had  especially 
developed. 

The  Barycentrischer  Calciil  in  the  year  1827  fur- 
nished the  first  example  of  homogeneous  co-ordinates, 
and  along  with  them  a  symmetry  in  the  developed 
formulae  hitherto  unknown  to  analytic  geometry.  In 
this  calculus  Mobius  started  with  the  assumption  that 
every  point  in  the  plane  of  a  triangle  ABC  may  be  re- 
garded as  the  center  of  gravity  of  the  triangle.  In 
this  case  there  belong  to  the  points  corresponding 
weights  which  are  exactly  the  homogeneous  co-ordi- 
nates of  the  point  P  with  respect  to  the  vertices  of 
the  fundamental  triangle  ABC.  By  means  of  this 
algorism  Mobius  found  by  algebraic  methods  a  series 
of  geometric  theorems,  for  example  those  expressing 
invariant  properties  like  the  theorems  on  cross-ratios. 
These  theorems,  found  analytically,  Mobius  sought  to 
demonstrate  geometrically  also,  and  for  this  purpose 
he  introduced  with  all  its  consequences  the  "law  of 
signs  "  which  expresses  that  for  A,  B,  C,  points  of  a 
straight  line,  AB  =  —  BA,  AB  +  BA  =  Q,  AB  +  BC 


Independently  of  M6bius,  but  starting  from  the  same  prin- 
ciples, Bellavitis  came  upon  his  new  geometric  method  of  equi- 
pollences.*  Two  equal  and  parallel  lines  drawn  in  the  same  direc- 
tion, AB  and  CD,  are  called  equipollent  (in  Cayley's  notation  AB 
^  CD).  By  this  assumption  the  whole  theory  is  reduced  to  the 

*  Bellavitis,  "  Saggio  di  Applicazioni  di  un  Nuovo  Metodo  di  Geometria 
Analitica  (Calcolo  delle  Equipollenze),"  in  Ann.  Lonib.  Veneto,  t.  5,  1835. 


GEOMETRY.  251 

consideration  of  segments  proceeding  from  a  ffxed  point.  Further 
it  is  assumed  that  AB  +  BC=AC  (Addition).  Finally  for  the  seg- 
ments a,  b,  c,  d,  with  inclinations  a,  ft,  y,  6  to  a  fixed  axis,  the 
equation  a=  —  must  not  only  be  a  relation  between  lengths  but 
must  also  show  that  <*=£  +  y—  &  (Proportion).  For  d=l  and 
a  =  0  this  becomes  a=bc,  i.  e.,  the  product  of  the  absolute  values 
of  the  lengths  is  a  =  bc  and  at  the  same  time  o  =  /3-f  y  (Multipli- 
cation). Equipollence  is  therefore  only  a  special  case  of  the  equal- 
ity of  two  objects,  applied  to  segments.* 

MObius  further  introduced  the  consideration  of 
correspondences  of  two  geometric  figures.  The  one- 
to-one  correspondence,  in  which  to  every  point  of  a 
first  figure  there  corresponds  one  and  only  one  point 
of  a  second  figure  and  to  every  point  of  the  second 
one  and  only  one  point  of  the  first,  Mobius  called  col- 
lineation.  He  constructed  not  only  a  collinear  image 
of  the  plane  but  also  of  ordinary  space. 

These  new  and  fundamental  ideas  which  Mobius 
had  laid  down  in  the  barycentric  calculus  remained 
for  a  long  time  almost  unheeded  and  hence  did  not  at 
once  enter  into  the  formation  of  geometric  concep- 
tions. The  works  of  Plucker  and  Steiner  found  a 
more  favorable  soil.  The  latter  "had  recognized  in 
immediate  geometric  perception  the  sufficient  means 
and  the  only  object  of  his  knowledge.  Plucker,  on 
the  other  hand,f  sought  his  proofs  in  the  identity  of 
the  analytic  operation  and  the  geometric  construc- 

*Stolz,  O.,  Vorlesvngen  uber  allgemeine  Arithmetik,  1885-1886. 

f'Clebsch,  Versuch  einer  Darlegung  und  Wiirdigung  seiner  wissenschaft- 
lichen  Leistungen  von  einigen  seiner  Freunde  (Brill,  Gordan,  Klein,  Luroth, 
A.  Mayer,  Nether,  Von  der  Miihll),"  in  Math.  Ann.,  Bd.  7. 


252  HISTORY  OF  MATHEMATICS. 

tion,  and  regarded  geometric  truth  only  as  one  of  the 
many  conceivable  antitypes  of  analytic  relation." 

At  a  later  period  (1855)  Mobius  engaged  in  the 
study  of  involutions  of  higher  degree.  Such  an  invo- 
lution of  the  tnth  degree  consists  of  two  groups  each 
of  m  points  :  A\,  A$,  As,  .  .  .  Am ;  B\,  B^,  Bz,  .  .  .  Bni, 
which  form  two  figures  in  such  a  way  that  to  the  1st, 
2d,  3d,  .  .  .  mih  points  of  one  group,  as  points  of  the 
first  figure,  there  correspond  in  succession  the  2d,  3d, 
4th  .  .  .  1st  points  of  the  same  group  as  points  of  the 
second  figure,  with  the  same  determinate  relation.  In- 
volutions of  higher  degree  had  been  previously  studied 
by  Poncelet  (1843).  He  started  from  the  theorem 
given  by  Sturm  (1826),  that  by  the  conic  sections  of 
the  surfaces  of  the  second  order  «  =  0,  v  =  Q,  u-\-Xv 
=  0,  there  are  determined  upon  a  straight  line  six 
points,  A,  A',  B,  B',  C,  C'  in  involution,  i.  e.,  so  that 
in  the  systems  ABCA'B'C'  and  A'B'C'ABC  not  only 
A  and  A',  B  and  B',  C  and  C',  but  also  A'  and  A,  B' 
and  B,  C'  and  C  are  corresponding  point-pairs.  This 
mutual  correspondence  of  three  point-pairs  of  a  line 
Desargues  had  already  (in  1639)  designated  by  the 
term  "involution."* 

Pliicker  is  the  real  founder  of  the  modern  analytic 
tendency,  and  he  attained  this  distinction  by  "formu- 
lating analytically  the  principle  of  duality  and  follow- 
ing out  its  consequences. ""}"  His  Aiialytisch-geometri- 
sche  Untersuchungen  appeared  in  1828.  By  this  work 

*Baltzer.  t  Brill,  A.,  Antrittsrede  in  Tubingen,  1884. 


GEOMETRY.  253 

was  created  for  geometry  the  method  of  symbolic  no- 
tation and  of  undetermined  coefficients,  whereby  one 
is  freed  from  the  necessity,  in  the  consideration  of  the 
mutual  relations  of  two  figures,  of  referring  to  the 
system  of  co-ordinates,  so  that  he  can  deal  with  the 
figures  themselves.  The  System  der  analytischen  Geo- 
metric of  1835  furnishes,  besides  the  abundant  appli- 
cation of  the  abbreviated  notation,  a  complete  classi- 
fication of  plane  curves  of  the  third  order.  In  the 
Theorie  der  algebraischen  Kurven  of  1839,  in  addition 
to  an  investigation  of  plane  curves  of  the  fourth  order 
there  appeared  those  analytic  relations  between  the 
ordinary  singularities  of  plane  curves  which  are  gen- 
erally known  as  "Plucker's  equations." 

These  Plucker  equations  which  at  first  are  applied 
only  to  the  four  dualistically  corresponding  singulari- 
ties (point  of  inflexion,  double  point,  inflexional  tan- 
gent, double  tangent)  were  extended  by  Cayley  to 
curves  with  higher  singularities.  By  the  aid  of  devel- 
opments in  series  he  derived  four  "equivalence  num- 
bers" which  enable  us  to  determine  how  many  singu- 
larities are  absorbed  into  a  singular  point  of  higher 
order,  and  how  the  expression  for  the  deficiency  of 
the  curve  is  modified  thereby.  Cayley's  results  were 
confirmed,  extended,  and  completed  as  to  proofs  by 
the  works  of  Nother,  Zeuthen,  Halphen,  and  Smith. 
The  fundamental  question  arising  from  the  Cayley 
method  of  considering  the  subject,  whether  and  by 
what  change  of  parameters  a  curve  with  correspond- 


254  HISTORY  OF  MATHEMATICS. 

ing  elementary  singularities  can  be  derived  from  a 
curve  with  higher  singularity,  for  which  the  Pliicker 
and  deficiency  equations  are  the  same,  has  been 
studied  by  A.  Brill. 

Plucker's  greatest  service  consisted  in  the  intro- 
duction of  the  straight  line  as  a  space  element.  The 
principle  of  duality  had  led  him  to  introduce,  besides 
the  point  in  the  plane,  the  straight  line,  and  in  space 
the  plane  as  a  determining  element.  Pliicker  also 
used  in  space  the  straight  line  for  the  systematic  gen- 
eration of  geometric  figures.  His  first  works  in  this 
direction  were  laid  before  the  Royal  Society  in  Lon- 
don in  1865.  They  contained  theorems  on  complexes, 
congruences,  and  ruled  surfaces  with  some  indications 
of  the  method  of  proof.  The  further  development 
appeared  in  1868  as  Neue  Geometric  des  Raumes,  ge- 
griindet  auf  die  Betrachtung  der  geraden  Linie  als  Raum- 
element.  Pliicker  had  himself  made  a  study  of  linear 
complexes  but  his  completion  of  the  theory  of  com- 
plexes of  the  second  degree  was  interrupted  by  death. 
Further  extension  of  the  theory  of  complexes  was 
made  by  F.  Klein. 

The  results  contained  in  Plucker's  last  work  have 
thrown  a  flood  of  light  upon  the  difference  between 
plane  and  solid  geometry.  The  curved  line  of  the 
plane  appears  as  a  simply  infinite  system  either  of 
points  or  of  straight  lines ;  in  space  the  curve  may  be 
regarded  as  a  simply  infinite  system  of  points,  straight 
lines  or  planes ;  but  from  another  point  of  view  this 


GEOMETRY.  255 

curve  in  space  may  be  replaced  by  the  developable 
surface  of  which  it  is  the  edge  of  regression.  Special 
cases  of  the  curve  in  space  and  the  developable  sur- 
face are  the  plane  curve  and  the  cone.  A  further 
space  figure,  the  general  surface,  is  on  the  one  side  a 
doubly  infinite  system  of  points  or  planes,  but  on  the 
other,  as  a  special  case  of  a  complex,  a  triply  infinite 
system  of  straight  lines,  the  tangents  to  the  surface. 
As  a  special  case  we  have  the  skew  surface  or  ruled 
surface.  Besides  this  the  congruence  appears  as  a 
doubly,  the  complex  as  a  triply,  infinite  system  of 
straight  lines.  The  geometry  of  space  involves  a  num- 
ber of  theories  to  which  plane  geometry  offers  no  anal- 
ogy. Here  belong  the  relations  of  a  space  curve  to 
the  surfaces  which  may  be  passed  through  it,  or  of  a 
surface  to  the  gauche  curves  lying  upon  it.  To  the 
lines  of  curvature  upon  a  surface  there  is  nothing 
corresponding  in  the  plane,  and  in  contrast  to  the 
consideration  of  the  straight  line  as  the  shortest  line 
between  two  points  of  a  plane,  there  stand  in  space 
two  comprehensive  and  difficult  theories,  that  of  the 
geodetic  line  upon  a  given  surface  and  that  of  the 
minimal  surface  with  a  given  boundary.  The  ques- 
tion of  the  analytic  representation  of  a  gauche  curve 
involves  peculiar  difficulties,  since  such  a  figure  can 
be  represented  by  two  equations  between  the  co-ordi- 
nates x,  y,  z  only  when  the  curve  is  the  complete  in- 
tersection of  two  surfaces.  In  just  this  direction  tend 


256  HISTORY  OF  MATHEMATICS. 

the  modern  investigations  of  Nother,  Halphen,  and 
Valentiner. 

Four  years  after  the  Analytisch-geometrische  Unter- 
suchungen  of  Plucker,  in  the  year  1832,  Steiner  pub- 
lished his  Systematische  Entwicklung  der  Abhdngigkeit 
geometrischer  Gestalten.  Steiner  found  the  whole  the- 
ory of  conic  sections  concentrated  in  the  single  theo- 
rem (with  its  dualistic  analogue)  that  a  curve  of  the 
second  order  is  produced  as  the  intersection  of  two 
collinear  or  projective  pencils,  and  hence  the  theory 
of  curves  and  surfaces  of  the  second  order  was  essen- 
tially completed  by  him,  so  that  attention  could  be 
turned  to  algebraic  curves  and  surfaces  of  higher  or- 
der. Steiner  himself  followed  this  course  with  good 
results.  This  is  shown  by  the  "Steiner  surface,"  and 
by  a  paper  which  appeared  in  1848  in  the  Berliner 
Abhandlungen.  In  this  the  theory  of  the  polar  of  a 
point  with  respect  to  a  curved  line  was  treated  ex- 
haustively and  thus  a  more  geometric  theory  of  plane 
curves  developed,  which  was  further  extended  by  the 
labors  of  Grassmann,  Chasles,  Jonquieres,  and  Cre- 
mona.* 

The  names  of  Steiner  and  Plucker  are  also  united  in  connec- 
tion with  a  problem  which  in  its  simplest  form  belongs  to  elemen- 
tary geometry,  but  in  its  generalization  passes  into  higher  fields. 
It  is  the  Malfatti  Problem. f  In  1803  Malfatti  gave  out  the  following 
problem:  From  a  right  triangular  prism  to  cut  out  three  cylinders 
which  shall  have  the  same  altitude  as  the  prism,  whose  volumes 
shall  be  the  greatest  possible,  and  consequently  the  mass  remain- 
»  Loria.  t  Wittstein,  Geschichte  des  Malfatti  'schen  Problems,  1871 . 


GEOMETRY.  257 

ing  after  their  removal  shall  be  a  minimum.  This  problem  he  re- 
duced to  what  is  now  generally  known  as  Malfatti's  problem  :  In  a 
given  triangle  to  inscribe  three  circles  so  that  each  circle  shall  be 
tangent  to  two  sides  of  the  triangle  and  to  the  other  two  circles.  He 
calculates  the  radii  xlt  xz,  x3  of  the  circles  sought  in  terms  of  the 
semi  perimeter  5  of  the  triangle,  th'e  radius  p  of  the  inscribed  cir- 
cle, the  distances  alt  a2,  a8  ;  bl>  bz,  ba  of  the  vertices  of  the  tri- 
angle from  the  center  of  the  inscribed  circle  and  its  points  of  tan- 
gency  to  the  sides,  and  gets  : 

X1  =  ~ 


*3  =     -  (s  +  «s  —  P  —  «i  —  «a). 

without  giving  the  calculation  in  full  ;  but  he  adds  a  simple  con- 
struction. Steiner  also  studied  this  problem.  He  gave  (without 
proof)  a  construction,  showed  that  there  are  thirty-two  solutions 
and  generalized  the  problem,  replacing  the  three  straight  lines  by 
three  circles.  Pliicker  also  considered  this  same  generalization. 
But  besides  this  Steiner  studied  the  same  problem  for  space  :  In 
connection  with  three  given  conies  upon  a  surface  of  the  second 
order  to  determine  three  others  which  shall  each  touch  two  of  the 
given  conies  and  two  of  the  required.  This  general  problem  re- 
ceived an  analytic  solution  from  Schellbach  and  Cayley,  and  also 
from  Clebsch  with  the  aid  of  the  addition  theorem  of  elliptic  func- 
tions, while  the  more  simple  problem  in  the  plane  was  attacked  in 
the  greatest  variety  of  ways  by  Gergonne,  Lehmus,  Crelle,  Grunert, 
Scheffler,  Schellbach  (who  gave  a  specially  elegant  trigonometric 
solution)  and  Zorer.  The  first  perfectly  satisfactory  proof  of  Stei- 
ner's  construction  was  given  by  Binder.* 

After  Steiner  came  von  Staudt  and  Chasles  who 
rendered  excellent  service  in  the  development  of  pro- 

*Programm  Schonthal,  1868. 


258  HISTORY  OF  MATHEMATICS. 

jective  geometry.  In  1837  Michel  Chasles  published 
his  Aper$u  historique  sur  Vorigine  et  le  dtveloppement 
des  mtthodes  en  ge'ome'trie,  a  work  in  which  both  ancient 
and  modern  methods  are  employed  in  the  derivation 
of  many  interesting  results,  of  which  several  of  the 
most  important,  among  them  the  introduction  of  the 
cross-ratio  (Chasles's  "anharmonic  ratio")  and  the 
reciprocal  and  collinear  relation  (Chasles's  "duality" 
and  "homography"),  are  to  be  assigned  in  part  to 
Steiner  and  in  part  to  Mobius. 

Von  Staudt's  Geometric  der  Lage  appeared  in  1847, 
his  Beitrdge  zur  Geometric  der  Lage,  1856-1860.  These 
works  form  a  marked  contrast  to  those  of  Steiner  and 
Chasles  who  deal  continually  with  metric  relations 
and  cross-ratios,  while  von  Staudt  seeks  to  solve  the 
problem  of  "making  the  geometry  of  position  an  in- 
dependent science  not  standing  in  need  of  measure- 
ment." Starting  from  relations  of  position  purely, 
von  Staudt  develops  all  theorems  that  do  not  deal 
immediately  with  the  magnitude  of  geometric  forms, 
completely  solving,  for  example,  the  problem  of  the 
introduction  of  the  imaginary  into  geometry.  The 
earlier  works  of  Poncelet,  Chasles,  and  others  had, 
to  be  sure,  made  use  of  complex  elements  but  had 
denned  the  same  in  a  manner  more  or  less  vague  and, 
for  example,  had  not  separated  conjugate  complex 
elements  from  each  other.  Von  Staudt  determined 
the  complex  elements  as  double  elements  of  involu- 
tion-relations. Each  double  element  is  characterized 


GEOMETRY.  259 

by  the  sense  in  which,  by  this  relation,  we  pass  from 
the  one  to  the  other.  This  suggestion  of  von  Staudt's, 
however,  did  not  become  generally  fruitful,  and  it 
was  reserved  for  later  works  to  make  it  more  widely 
known  by  the  extension  of  the  originally  narrow  con- 
ception. 

In  the  Beitrdge  von  Staudt  has  also  shown  how 
the  cross-ratios  of  any  four  elements  of  a  prime  form 
of  the  first  class  (von  Staudt's  Wiirfe)  may  be  used  to 
derive  absolute  numbers  from  pure  geometry.* 

With  the  projective  geometry  is  most  closely  con- 
nected the  modern  descriptive  geometry.  The  former 
in  its  development  drew  its  first  strength  from  the 
considerations  of  perspective,  the  latter  enriches  itself 
later  with  the  fruits  matured  by  the  cultivation  of  pro- 
jective geometry. 

The  perspective  of  the  Renaissance  f  was  devel- 
oped especially  by  French  mathematicians,  first  by 
Desargues  who  used  co-ordinates  in  his  pictorial  rep- 
resentation of  objects  in  such  a  way  that  two  axes  lay 
in  the  picture  plane,  while  the  third  axis  was  normal 
to  this  plane.  The  results  of  Desargues  were  more 
important,  however,  for  theory  than  for  practice. 
More  valuable  results  were  secured  by  Taylor  with  a 
"linear  perspective"  (1715).  In  this  a  straight  line 
is  determined  by  its  trace  and  vanishing  point,  a  plane 
by  its  trace  and  vanishing  line.  This  method  was 

*  Stolz,  O.,  Vorlesungen  iiber  allgemeinc  Arzthmetik,  1885-1886. 
t  Wiener. 


260  HISTORY  OF  MATHEMATICS. 

used  by  Lambert  in  an  ingenious  manner  for  different 
c  instructions,  so  that  by  the  middle  of  the  eighteenth 
century  even  space-forms  in  general  position  could  be 
pictured  in  perspective. 

Out  of  the  perspective  of  the  eighteenth  century 
grew  "descriptive  geometry,"  first  in  a  work  of  Fre"- 
zier's,  which  besides  practical  methods  contained  a 
special  theoretical  section  furnishing  proofs  for  all 
cases  of  the  graphic  methods  considered.  Even  in 
the  "description,"  or  representation,  Fre"zier  replaces 
the  central  projection  by  the  perpendicular  parallel- 
projection,  "which  may  be  illustrated  by  falling  drops 
of  ink."*  The  picture  of  the  plane  of  projection  is 
called  the  ground  plane  or  elevation  according  as  the 
picture  plane  is  horizontal  or  vertical.  With  the  aid 
of  this  "description"  Frezier  represents  planes,  poly- 
hedra,  surfaces  of  the  second  degree  as  well  as  inter- 
sections and  developments. 

Since  the  time  of  Monge  descriptive  geometry  has 
taken  rank  as  a  distinct  science.  The  Lemons  de  geo- 
mttrie  descriptive  (1795)  form  the  foundation-pillars  of 
descriptive  geometry,  since  they  introduce  horizontal 
and  vertical  planes  with  the  ground-line  and  show 
how  to  represent  points  and  straight  lines  by  two  pro- 
jections, and  planes  by  two  traces.  This  is  followed 
in  the  Lemons  by  the  great  number  of  problems  of  in- 
tersection, contact  and  penetration  which  arise  from 
combinations  of  planes  with  polyhedra  and  surface = 


GEOMETRY.  261 

of  the  second  order.  Monge's  successors,  Lacroix, 
Hachette,  Olivier,  and  J.  de  la  Gournerie  applied 
these  methods  to  surfaces  of  the  second  order,  ruled 
surfaces,  and  the  relations  of  curvature  of  curves  and 
surfaces. 

Just  at  this  time,  when  the  development  of  descriptive  geom- 
etry in  France  had  borne  its  first  remarkable  results,  the  technical 
high  schools  came  into  existence.  In  the  year  1794  was  established 
in  Paris  the  £cole  Centrale  des  Travaux  Publics  from  which  in 
1795  the  £cole  Polytechnique  was  an  outgrowth.  Further  techni- 
cal schools,  which  in  course  of  time  attained  to  university  rank, 
were  founded  in  Prague  in  1806,  in  Vienna  in  1815,  in  Berlin  in 
1820,  in  Karlsruhe  in  1825,  in  Munich  in  1827,  in  Dresden  in  1828, 
in  Hanover  in  1831,  in  Stuttgart  1832,  in  Zurich  in  1860,  in 
Braunschweig  in  1862,  in  Darmstadt  in  1869,  and  in  Aix-la-Chapelle 
in  1870.  In  these  institutions  the  results  of  projective  geometry 
were  used  to  the  greatest  advantage  in  the  advancement  of  descrip- 
tive geometry,  and  were  set  forth  in  the  most  logical  manner  by 
Fiedler,  whose  text-books  and  manuals,  in  part  original  and  in 
part  translations  from  the  English,  take  a  conspicuous  place  in  the 
literature  of  the  science. 

With  the  technical  significance  of  descriptive  geometry  there 
has  been  closely  related  for  some  years  an  artistic  side,  and  it  is 
this  especially  which  has  marked  an  advance  in  works  on  axonom- 
etry  (Weisbach,  1844),  relief-perspective,  photogrammetry,  and 
theory  of  lighting. 

The  second  quarter  of  our  century  marks  the  time 
when  developments  in  form-theory  in  connection  with 
geometric  constructions  have  led  to  the  discovery  of 
of  new  and  important  results.  Stimulated  on  the  one 
side  by  Jacobi,  on  the  other  by  Poncelet  and  Steiner, 


262  HISTORY  OF  MATHEMATICS. 

Hesse  (1837-1842)  by  an  application  of  the  transfor- 
mation of  homogeneous  forms  treated  the  theory  of 
surfaces  of  the  second  order  and  constructed  their 
principal  axes.*  By  him  the  notions  of  "polar  tri- 
angles" and  "polar  tetrahedra"  and  of  "systems  of 
conjugate  points"  were  introduced  as  the  geometric 
expression  of  analytic  relations.  To  these  were  added 
the  linear  construction  of  the  eighth  intersection  of 
three  surfaces  of  the  second  degree,  when  seven  of 
them  are  given,  and  also  by  the  use  of  Steiner's  theo- 
rems, the  linear  construction  of  a  surface  of  the  sec 
ond  degree  from  nine  given  points.  Clebsch,  follow 
ing  the  English  mathematicians,  Sylvester,  Cayley, 
and  Salmon,  went  in  his  works  essentially  further  than 
Hesse.  His  vast  contributions  to  the  theory  of  in- 
variants, his  introduction  of  the  notion  of  the  defi- 
ciency of  a  curve,  his  applications  of  the  theory  of 
elliptic  and  Abelian  functions  to  geometry  and  to  the 
study  of  rational  and  elliptic  curves,  secure  for  him  a 
pre-eminent  place  among  those  who  have  advanced 
the  science  of  extension.  As  an  algebraic  instrument 
Clebsch,  like  Hesse,  had  a  fondness  for  the  theorem 
upon  the  multiplication  of  determinants  in  its  appli- 
cation to  bordered  determinants.  His  worksf  upon 
the  general  theory  of  algebraic  curves  and  surfaces 


* N8ther,  "Otto  Hesse,"  Schlomilch's Zeitschrift,  Bd.  20,  HI.  A. 

t"  Clebsch,  Versuch  einer  Darlegung  und  Wurdigung  seiner  wissen- 
schaftlichen  Leistungen  von  einigen  seiner  Freunde  "  (Brill,  Gordan,  Klein, 
Liiroth,  A.  Mayer,  Nother,  Von  der  Miihll)  Math.  Ann.,  Bd.  7. 


GEOMETRY.  263 

began  with  the  determination  of  those  points  upon  an 
algebraic  surface  at  which  a  straight  line  has  four- 
point  contact,  a  problem  also  treated  by  Salmon  but 
not  so  thoroughly.  While  now  the  theory  of  surfaces 
of  the  third  order  with  their  systems  of  twenty-seven 
straight  lines  was  making  headway  on  English  soil, 
Clebsch  undertook  to  render  the  notion  of  "defi- 
ciency" fruitful  for  geometry.  This  notion,  whose 
analytic  properties  were  not  unknown  to  Abel,  is  found 
first  in  Riemann's  Theorie  der  Abel'schen  Funktionen 
(1857).  Clebsch  speaks  also  of  the  deficiency  of  an 
algebraic  curve  of  the  «th  order  with  d  double  points 
and  r  points  of  inflexion,  and  determines  the  number 
p  =  %(n  —  !)(«  — 2)  —  d — r.  To  one  class  of  plane 
or  gauche  curves  characterized  by  a  definite  value  of 
p  belong  all  those  that  can  be  made  to  pass  over  into 
one  another  by  a  rational  transformation  or  which 
possess  the  property  that  any  two  have  a  one-to-one 
correspondence.  Hence  follows  the  theorem  that  only 
those  curves  that  possess  the  same  3/ — 3  parameters 
(for  curves  of  the  third  order,  the  same  one  parame- 
ter) can  be  rationally  transformed  into  one  another. 

The  difficult  theory  of  gauche  curves*  owes  its  first 
general  results  to  Cayley,  who  obtained  formulae  cor- 
responding to  Plucker's  equations  for  plane  curves. 
Works  on  gauche  curves  of  the  third  and  fourth  orders 
had  already  been  published  by  Mobius,  Chasles,  and 
Von  Staudt.  General  observations  on  gauche  curves 


264  HISTORY  OF  MATHEMATICS. 

in  more  recent  times  are  found  in  theorems  of  Nether 
and  Halphen. 

The  foundations  of  enumerative  geometry*  are 
found  in  Chasles's  method  of  characteristics  (1864). 
Chasles  determined  for  rational  configurations  of  one 
dimension  a  correspondence-formula  which  in  the 
simplest  case  may  be  stated  as  follows  :  If  two  ranges 
of  points  R\  and  RI  lie  upon  a  straight  line  so  that  to 
every  point  x  of  R\  there  correspond  in  general  a 
points  y  in  RI,  and  again  to  every  point  y  of  RI  there 
always  correspond  ft  points  x  in  J?i,  the  configuration 
formed  from  R\  and  R^  has  (a-|-/3)  coincidences  or 
there  are  (a  +  /?)  times  in  which  a  point  x  coincides 
with  a  corresponding  point  y.  The  Chasles  corre 
spondence-principle  was  extended  inductively  by  Cay- 
ley  in  1866  to  point-systems  of  a  curve  of  higher 
deficiency  and  this  extension  was  proved  by  Brill,  f 
Important  extensions  of  these  enumerative  formulae 
(correspondence-formulae),  relating  to  general  alge- 
braic curves,  have  been  given  by  Brill,  Zeuthen,  and 
Hurwitz,  and  set  forth  in  elegant  form  by  the  intro- 
duction of  the  notion  of  deficiency.  An  extended 
treatment  of  the  fundamental  problem  of  enumerative 
geometry,  to  determine  how  many  geometric  config- 
urations of  given  definition  satisfy  a  sufficient  number 
of  conditions,  is  contained  in  the  Kalkiil  der  abzdhlen- 
den  Geometric  by  H.  Schubert  (1879). 

The  simplest  cases  of  one-to-one   correspondence 

*Loria.  t  Mathcm .  Annalen,  VI. 


GEOMETRY.  265 

or  uniform  representation,  are  furnished  by  two  planes 
superimposed  one  upon  the  other.  These  are  the 
similarity  studied  by  Poncelet  and  the  collineation 
treated  by  Mobius,  Magnus,  and  Chasles.*  In  both 
cases  to  a  point  corresponds  a  point,  to  a  straight  line 
a  straight  line.  From  these  linear  transformations 
Poncelet,  Plucker,  Magnus,  Steiner  passed  to  the 
quadratic  where  they  first  investigated  one-to-one  cor- 
respondences between  two  separate  planes.  The 
"Steiner  projection"  (1832)  employed  two  planes  JE\ 
and  EI  together  with  two  straight  lines  gi  and  gi  not 
co-planar.  If  we  draw  through  a  point  P\  or  /*2  of  E\ 
or  EI  the  straight  line  #1  or  x%  which  cuts  g\  as  well 
as  g%,  and  determines  the  intersection  X%  or  X\,  with 
E%  or  JSi,  then  are  P\  and  X%,  and  PI  and  X\  corre- 
sponding points.  In  this  manner  to  every  straight 
line  of  the  one  plane  corresponds  a  conic  section  in 
the  other.  In  1847  Plucker  had  determined  a  point 
upon  the  hyperboloid  of  one  sheet,  like  fixing  a  point 
in  the  plane,  by  the  segments  cut  off  upon  the  two 
generators  passing  through  the  point  by  two  fixed 
generators.  This  was  an  example  of  a  uniform  rep- 
resentation of  a  surface  of  the  second  order  upon  the 
plane. 

The  one-to-one  relation  of  an  arbitrary  surface  of 
the  second  order  to  the  plane  was  investigated  by 
Chasles  in  1863,  and  this  work  marks  the  beginning 
of  the  proper  theory  of  surface  representation  which 


266  HISTORY  OF  MATHEMATICS. 

found  its  further  development  when  Clebsch  and  Cre 
mona  independently  succeeded  in  the  representation 
of  surfaces  of  the  third  order.  Cremona's  important 
results  were  extended  by  Cayley,  Clebsch,  Rosanes, 
and  Nother,  to  the  last  of  whom  we  owe  the  impor- 
tant theorem  that  every  Cremona  transformation  which 
as  such  is  uniform  forward  and  backward  can  be 
effected  by  the  repetition  of  a  number  of  quadratic 
transformations.  In  the  plane  only  is  the  aggregate 
of  all  rational  or  Cremona  transformations  known  ; 
for  the  space  of  three  dimensions,  merely  a  beginning 
of  the  development  of  this  theory  has  been  made.* 

A  specially  important  case  of  one-to-one  corre- 
spondence is  that  of  a  conformal  representation  of  a 
surface  upon  the  plane,  because  here  similarity  in  the 
smallest  parts  exists  between  original  and  image.  The 
simplest  case,  the  stereographic  projection,  was  known 
to  Hipparchus  and  Ptolemy.  The  representation  by 
reciprocal  radii  characterized  by  the  fact  that  any  two 
corresponding  points  P\  and  P*  lie  upon  a  ray  through 
the  fixed  point  O  so  that  OP\  •  OP9  =  constant,  is  also 
conformal.  Here  every  sphere  in  space  is  in  general 
transformed  into  a  sphere.  This  transformation,  stud- 
ied by  Bellavitis  1836  and  Stubbs  1843,  is  especially 
useful  in  dealing  with  questions  of  mathematical  phys- 
ics. Sir  Wm.  Thomson  calls  it  "the  principle  of  elec- 
tric images."  The  investigations  upon  representa- 


*  Klein,  F.,  Vergleichende  Betrachtungen  &ber  neuere  geometrische  Forsch- 
ngen,  1872. 


GEOMETRY.  267 

tions,  made  by  Lambert  and  Lagrange,  but  more 
especially  those  by  Gauss,  lead  to  the  theory  of  curva- 
ture. 

A  further  branch  of  geometry,  the  differential  ge- 
ometry (theory  of  curvature  of  surfaces),  considers  in 
general  not  first  the  surface  in  its  totality  but  the 
properties  of  the  same  in  the  neighborhood  of  an  or- 
dinary point  of  the  surface,  and  with  the  aid  of  the 
differential  calculus  seeks  to  characterize  it  by  ana- 
lytic formulae. 

The  first  attempts  to  enter  this  domain  were  made 
by  Lagrange  (1761),  Euler  (1766),  and  Meusnier(1776). 
The  former  determined  the  differential  equation  of 
minimal  surfaces ;  the  two  latter  discovered  certain 
theorems  upon  radii  of  curvature  and  surfaces  of  cen- 
ters. But  of  fundamental  importance  for  this  rich  do- 
main have  been  the  investigations  of  Monge,  Dupin, 
and  especially  of  Gauss.  In  the  Application  de  I'ana- 
lyse  a  la  gtomttrie  (1795),  Monge  discusses  families 
of  surfaces  (cylindrical  surfaces,  conical  surfaces,  and 
surfaces  of  revolution, — envelopes  with  the  new  no- 
tions of  characteristic  and  edge  of  regression)  and  de- 
termines the  partial  differential  equations  distinguish- 
ing each.  In  the  year  1813  appeared  the  Dtveloppements 
de  gtomttrie  by  Dupin.  It  introduced  the  indicatrix 
at  a  point  of  a  surface,  as  well  as  extensions  of  the 
theory  of  lines  of  curvature  (introduced  by  Monge) 
and  of  asymptotic  curves. 

Gauss  devoted  to  differential  geometry  three  trea- 


268  HISTORY  OF  MATHEMATICS. 

tises :  the  most  celebrated,  Disquisitiones  generates  circa 
superficies  curvas,  appeared  in  1827,  the  other  two 
Untersuchungen  iiber  Gegenstdnde  der  hoheren  Geoddsie 
were  published  in  1843  and  1846.  In  the  Disquisi- 
tiones, to  the  preparation  of  which  he  was  led  by  his 
own  astronomical  and  geodetic  investigations,*  the 
spherical  representation  of  a  surface  is  introduced. 
The  one-to-one  correspondence  between  the  surface 
and  the  sphere  is  established  by  regarding  as  corre- 
sponding points  the  feet  of  parallel  normals,  where 
obviously  we  must  restrict  ourselves  to  a  portion  of 
the  given  surface,  if  the  correspondence  is  to  be  main- 
tained. Thence  follows  the  introduction  of  the  curvi- 
linear co-ordinates  of  a  surface,  and  the  definition  of 
the  measure  of  curvature  as  the  reciprocal  of  the  pro- 
duct of  the  two  radii  of  principal  curvature  at  the 
point  under  consideration.  The  measure  of  curvature 
is  first  determined  in  ordinary  rectangular  co-ordinates 
and  afterwards  also  in  curvilinear  co-ordinates  of  the 
surface.  Of  the  latter  expression  it  is  shown  that  it  is 
not  changed  by  any  bending  of  the  surface  without 
stretching  or  folding  (that  it  is  an  invariant  of  curva- 
ture). Here  belong  the  consideration  of  geodetic 
lines,  the  definition  and  a  fundamental  theorem  upon 
the  total  curvature  (curvatura  Integra)  of  a  triangle 
bounded  by  geodetic  lines. 

The  broad  views  set  forth  in  the  Disquisitiones  of 
1827  sent  out  fruitful  suggestions  in  the  most  vari- 

*  Brill,  A.,  Antrittsrede  in  Tubingen,  1884. 


GEOMETRY.  269 

ous  directions.  Jacobi  determined  the  geodetic  lines 
of  the  general  ellipsoid.  With  the  aid  of  elliptic  co- 
ordinates (the  parameters  of  three  surfaces  of  a  sys- 
tem of  confocal  surfaces  of  the  second  order  passing 
through  the  point  to  be  determined)  he  succeeded  in 
integrating  the  partial  differential  equation  so  that  the 
equation  of  the  geodetic  line  appeared  as  a  relation 
between  two  Abelian  integrals.  The  properties  of  the 
geodetic  lines  of  the  ellipsoid  are  derived  with  espe- 
cial ease  from  the  elegant  formulae  given  by  Liou- 
ville.  By  Lame*  the  theory  of  curvilinear  co-ordinates, 
of  which  he  had  investigated  a  special  case  in  1837, 
was  developed  in  1859  into  a  theory  for  space  in  his 
Lemons  sur  la  thtorie  des  coordonntes  turvilignes. 

The  expression  for  the  Gaussian  measure  of  curva- 
ture as  a  function  of  curvilinear  co-ordinates  has  given 
an  impetus  to  the  study  of  the  so-called  differential 
invariants  or  differential  parameters.  These  are  cer- 
tain functions  of  the  partial  derivatives  of  the  coeffi- 
cients in  the  expression  for  the  square  of  the  line- ele- 
ment which  in  the  transformation  of  variables  behave 
like  the  invariants  of  modern  algebra.  Here  Sauc6, 
Jacobi,  C.  Neumann,  and  Halphen  laid  the  founda- 
tions, and  a  general  theory  has  been  developed  by 
Beltrami.*  This  theory,  as  well  as  the  contact-trans- 
formations of  Lie,  moves  along  the  border  line  be- 
tween geometry  and  the  theory  of  differential  equa- 
tions, f 

•  Mem.  di  Bologna,  VIII.  t  Loria. 


270  HISTORY  OF  MATHEMATICS. 

With  problems  of  the  mathematical  theory  of  light  are  con- 
nected certain  investigations  upon  systems  of  rays  and  the  prop- 
erties of  infinitely  thin  bundles  of  rays,  as  first  carried  on  by  Du- 
pin,  Malus,  Ch.  Sturm,  Bertrand,  Transon,  and  Hamilton.  The 
celebrated  works  of  Kummer  (1857  and  1866)  perfect  Hamilton's 
results  upon  bundles  of  rays  and  consider  the  number  of  singular- 
ities of  a  system  of  rays  and  its  focal  surface.  An  interesting  ap- 
plication to  the  investigation  of  the  bundles  of  rays  between  the 
lens  and  the  retina,  founded  on  the  study  of  the  infinitely  thin 
bundles  of  normals  of  the  ellipsoid,  was  given  by  O.  Boklen.* 

Non-  Euclidean  Geometry. —  Though  the  respect 
which  century  after  century  had  paid  to  the  Elements 
of  Euclid  was  unbounded,  yet  mathematical  acuteness 
had  discovered  a  vulnerable  point;  and  this  point  f 
forms  the  eleventh  axiom  (according  to  Hankel,  reck- 
oned by  Euclid  himself  among  the  postulates)  which 
affirms  that  two  straight  lines  intersect  on  that  side  of 
a  transversal  on  which  the  sum  of  the  interior  angles 
is  less  than  two  right  angles.  Toward  the  end  of  the 
last  century  Legendre  had  tried  to  do  away  with  this 
axiom  by  making  its  proof  depend  upon  the  others,  but 
his  conclusions  were  invalid.  This  effort  of  Legendre's 
was  an  indication  of  the  search  now  beginning  after  a 
geometry  free  from  contradictions,  a  hyper-Euclidean 
geometry  or  pangeometry.  Here  also  Gauss  was 
among  the  first  who  recognized  that  this  axiom  could 
not  be  proved.  Although  from  his  correspondence 
with  Wolfgang  Bolyai  and  Schumacher  it  can  easily 

*  Knmecker't  Journal,  Band  46.    Forischritte,  1884. 
t  Lori  a. 


GEOMETRY.  271 

be  seen  that  he  had  obtained  some  definite  results  in 
this  field  at  an  early  period,  he  was  unable  to  decide 
upon  any  further  publication.  The  real  pioneers  in 
the  Non-Euclidean  geometry  were  Lobachevski  and 
the  two  Bolyais.  Reports  of  the  investigations  of 
Lobachevski  first  appeared  in  the  Courier  of  Kasan, 
1829-1830,  then  in  the  transactions  of  the  Univer- 
sity of  Kasan,  1835-1839,  and  finally  as  Geometrische 
Untcrsuchungen  iiber  die  Theorie  der  Parallellinien,  1840, 
in  Berlin.  By  Wolfgang  Bolyai  was  published  (1832- 
1833*)  a  two-volume  work,  Tentamen  Juventutem  stu- 
diosam  in  elementa  Matheseos  purae,  etc.  Both  works 
were  for  the  mathematical  world  a  long  time  as  good 
as  non-existent  till  first  Riemann,  and  then  (in  1866) 
R.  Baltzer  in  his  Elemente,  referred  to  Bolyai.  Almost 
at  the  same  time  there  followed  a  sudden  mighty  ad- 
vance toward  the  exploration  of  this  "new  world"  by 
Riemann,  Helmholtz,  and  Beltrami.  It  was  recog- 
nized that  of  the  twelve  Euclidean  axioms  f  nine  are 
of  essentially  arithmetic  character  and  therefore  hold 
for  every  kind  of  geometry ;  also  to  every  geometry  is 
applicable  the  tenth  axiom  upon  the  equality  of  all 
right  angles.  The  twelfth  axiom  (two  straight  lines, 
or  more  generally  two  geodetic  lines,  include  no 
space)  does  not  hold  for  geometry  on  the  sphere. 
The  eleventh  axiom  (two  straight  lines,  geodetic 


*  Schmidt,  "Aus  dem  Leben  zweier  nngarischen  Mathematiker,"  Grunert 
Arch.,  Bd.  48. 

t  Brill,  A.,  Ueber  das  elfte  Axiom  des  Euclid,  1883. 


272  HISTORY  OF  MATHEMATICS. 

lines,  intersect  when  the  sum  of  the  interior  angles  is 
less  than  two  right  angles)  does  not  hold  for  geometry 
on  a  pseudo-sphere,  but  only  for  that  in  the  plane. 

Riemann,  in  his  paper  "Ueber  die  Hypothesen, 
welche  der  Geometrie  zu  Grunde  liegen,"*  seeks  to 
penetrate  the  subject  by  forming  the  notion  of  a  mul- 
tiply extended  manifoldness ;  and  according  to  these 
investigations  the  essential  characteristics  of  an  «-ply 
extended  manifoldness  of  constant  measure  of  curva- 
ture are  the  following : 

1.  "Every  point  in  it  may  be  determined  by  n 
variable  magnitudes  (co-ordinates). 

2.  "The  length  of  a  line  is  independent  of  posi- 
tion and  direction,  so  that  ever)'  line  is  measurable 
by  every  other. 

3.  "To  investigate  the  measure-relations  in  such 
a  manifoldness,  we  must  for  every  point  represent  the 
line-elements  proceeding  from  it  by  the  corresponding 
differentials  of  the  co-ordinates.   This  is  done  by  virtue 
of  the  hypothesis  that  the  length-element  of  the  line 
is  equal  to  the  square  root  of  a  homogeneous  function 
of  the  second  degree  of  the  differentials  of  the  co- 
ordinates." 

At  the  same  time  Helmholtzf  published  in  the 
"Thatsachen,  welche  der  Geometrie  zu  Grunde  lie 
gen,"  the  following  postulates  : 

*  GSttinger  Abhandlungen,  XIII.,  1868.     Fortschritte,  1868. 
tFortschriite,  1868. 


GEOMETRY.  273 

1.  "A  point  of  an  n-tuple  manifoldness  is  deter- 
mined by  n  co-ordinates. 

2.  "Between  the  2n  co-ordinates  of  a  point-pair 
there  exists  an  equation,  independent  of  the  move- 
ment of  the  latter,  which  is  the  same  for  all  congruent 
point-pairs. 

3.  "Perfect  mobility  of  rigid  bodies  is  assumed. 

4.  "  If  a  rigid  body  of  n  dimensions  revolves  about 
n  —  1   fixed  points,  then  revolution  without  reversal 
will  bring  it  back  to  its  original  position." 

Here  spatial  geometry  has  satisfactory  foundations 
for  a  development  free  from  contradictions,  if  it  is  fur- 
ther assumed  that  space  has  three  dimensions  and  is 
of  unlimited  extent. 

One  of  the  most  surprising  results  of  modern  geo- 
metric investigations  was  the  proof  of  the  applicabil- 
ity of  the  non-Euclidean  geometry  to  pseudo-spheres 
or  surfaces  of  constant  negative  curvature.*  On  a 
pseudo-sphere,  for  example,  it  is  true  that  a  geodetic 
line  (corresponding  to  the  straight  line  in  the  plane, 
the  great  circle  on  the  sphere)  has  two  separate  points 
at  infinity;  that  through  a  point  P,  to  a  given  geodetic 
line  g,  there  are  two  parallel  geodetic  lines,  of  which, 
however,  only  one  branch  beginning  at  P  cuts  g  at  in- 
finity while  the  other  branch  does  not  meet  g  at  all ; 
that  the  sum  of  the  angles  of  a  geodetic  triangle  is 
less  than  two  right  angles.  Thus  we  have  a  geometry 
upon  the  pseudo-sphere  which  with  the  spherical  ge- 

*  Cayley,  Address  to  the  British  Association,  etc.,  1883. 


274  HISTORY  OF  MATHEMATICS. 

ometry  has  a  common  limiting  case  in  the  ordinary 
or  Euclidean  geometry.  These  three  geometries  have 
this  in  common  that  they  hold  for  surfaces  of  constant 
curvature.  According  as  the  constant  value  of  the 
curvature  is  positive,  zero,  or  negative,  we  have  to  do 
with  spherical,  Euclidean,  or  pseudo-spherical  geom- 
etry. 

A  new  presentation  of  the  same  theory  is  due  to 
F.  Klein.  After  projective  geometry  had  shown  that 
in  projection  or  linear  transformation  all  descriptive 
properties  and  also  some  metric  relations  of  the  fig- 
ures remain  unaltered,  the  endeavor  was  made  to  find 
for  the  metric  properties  an  expression  which  should 
remain  invariant  after  a  linear  transformation.  After 
a  preparatory  work  of  Laguerre  which  made  the  "no- 
tion of  the  angle  projective,"  Cayley,  in  1859,  found  the 
general  solution  of  this  problem  by  considering  "every 
metric  property  of  a  plane  figure  as  contained  in  a 
projective  relation  between  it  and  a  fixed  conic." 
Starting  from  the  Cayley  theory,  on  the  basis  of  the 
consideration  of  measurements  in  space,  Klein  suc- 
ceeded in  showing  that  from  the  projective  geometry 
with  special  determination  of  measurements  in  the 
plane  there  could  be  derived  an  elliptic,  parabolic, 
or  hyperbolic  geometry,*  the  same  fundamentally  as 
the  spherical,  Euclidean,  or  pseudo-spherical  geom- 
etry respectively. 

The  need  of  the  greatest  possible  generalization 

*  Fortsckritte,  1871. 


GEOMETRY.  275 

and  the  continued  perfection  of  the  analytic  apparatus 
have  led  to  the  attempt  to  build  up  a  geometry  of  n 
dimensions;  in  this,  however,  only  individual  relations 
have  been  considered.  Lagrange*  observes  that  "me- 
chanics may  be  regarded  as  a  geometry  of  four  dimen- 
sions." Plucker  endeavored  to  clothe  the  notion  of 
arbitrarily  extended  space  in  a  form  easily  understood. 
He  showed  that  for  the  point,  the  straight  line  or  the 
sphere,  the  surface  of  the  second  order,  as  a  space 
element,  the  space  chosen  must  have  three,  four,  or 
nine  dimensions  respectively.  The  first  investigation, 
giving  a  different  conception  from  Pliicker's  and  "con- 
sidering the  element  of  the  arbitrarily  extended  mani- 
foldness  as  an  analogue  of  the  point  of  space,"  is 
foundf  in  H.  Grassmann's  principal  work,  Die  Wissen- 
schaft  der  extensiven  Gross e  oder  die  lineale  Ausdehnungs- 
lehre  (1844),  which  remained  almost  wholly  unno- 
ticed, as  did  his  Geometrische  Analyse  (1847).  Then 
followed  Riemann's  studies  in  multiply  extended  mani- 
foldnesses  in  his  paper  Ueber  die  Hypothesen,  etc.,  and 
they  again  furnished  the  starting  point  for  a  series  of 
modern  works  by  Veronese,  H.  Schubert,  F.  Meyer, 
Segre,  Castelnuovo,  etc. 

A  Geometria  situs  in  the  broader  sense  was  created 
by  Gauss,  at  least  in  name;  but  of  it  we  know  scarcely 
more  than  certain  experimental  truths.  \  The  Analysis 

*  Loria. 

t  F.  Klein,  Vergleichende  Betrachtungeu  iiber  neuere  geometrische  For- 
s  hungen,  1872. 

t  Brill,  A.,  Antrittsrede  in  Tubingen,  1884. 


276  HISTORY  OF  MATHEMATICS. 

situs,  suggested  by  Riemann,  seeks  what  remains  fixed" 
after  transformations  consisting  of  the  combination  of 
infinitesimal  distortions.*  This  aids  in  the  solution 
of  problems  in  the  theory  of  functions.  The  contact 
transformations  already  considered  by  Jacobi  have 
been  developed  by  Lie.  A  contact-transformation  is 
defined  analytically  by  every  substitution  which  ex- 
presses the  values  of  the  co-ordinates  x,  y,  z,  and  the 

partial  derivatives  -7-  =p,  —r  =<?>  in  terms  of  quan- 
go: dy 

tities  of  the  same  kind,  x',  y',  z',  p',  tf.  In  such  a 
transformation  contacts  of  two  figures  are  replaced  by 
similar  contacts. 

Also  a  "geometric  theory  of  probability"  has  been 
created  by  Sylvester  and  Woolhousejf  Crofton  uses 
it  for  the  theory  of  lines  drawn  at  random  in  space. 

In  a  history  of  elementary  mathematics  there  pos- 
sibly calls  for  attention  a  related  field,  which  certainly 
cannot  be  regarded  as  a  branch  of  science,  but  yet 
which  to  a  certain  extent  reflects  the  development  of 
geometric  science,  the  history  of  geometric  illustrative 
material.J  Good  diagrams  or  models  of  systems  of 
space-elements  assist  in  teaching  and  have  frequently 
led  to  the  rapid  spread  of  new  ideas.  In  fact  in  the 
geometric  works  of  Euler,  Newton,  and  Cramer  are 
found  numerous  plates  of  figures.  Interest  in  the 

*  F.Klein.  1  Fortschritte,  1868. 

%  Brill,  A.,  Utber  die  Modcllsammlung  des  mathematischen  Seminars  der 
Unrversriat  Tubingen,  1886.  Mathtmatisch-naturwissenschaftliche  Mitthei- 
lungen  von  O.  Boklen.  1887. 


GEOMETRY.  277 

construction  of  models  seems  to  have  been  manifested 
first  in  France  in  consequence  of  the  example  and  ac- 
tivity of  Monge.  In  the  year  1830  the  Conservatoire 
des  arts  et  metiers  in  Paris  possessed  a  whole  series  of 
thread  models  of  surfaces  of  the  second  degree,  con- 
oids and  screw  surfaces.  A  further  advance  was  made 
by  Bardin  (1855).  He  had  plaster  and  thread  mod- 
els constructed  for  the  explanation  of  stone-cutting, 
toothed  gears  and  other  matters.  His  collection  was 
considerably  enlarged  by  Muret.  These  works  of 
French  technologists  met  with  little  acceptance  from 
the  mathematicians  of  that  country,  but,  on  the  con- 
trary, in  England  Cayley  and  Henrici  put  on  exhibi- 
tion in  London  in  1876  independently  constructed 
models  together  with  other  scientific  apparatus  of  the 
universities  of  London  and  Cambridge. 

In  Germany  the  construction  of  models  experi- 
enced an  advance  from  the  time  when  the  methods  of 
projective  geometry  were  introduced  into  descriptive 
geometry.  Plucker,  who  in  his  drawings  of  curves  of 
the  third  order  had  in  1835  showed  his  interest  in  re 
lations  of  form,  brought  together  in  1868  the  first 
large  collection  of  models.  This  consisted  of  models 
of  complex  surfaces  of  the  fourth  order  and  was  con- 
siderably enlarged  by  Klein  in  the  same  field.  A 
special  surface  of  the  fourth  order,  the  wave-surface 
for  optical  bi-axial  crystals  was  constructed  in  1840 
by  Magnus  in  Berlin,  and  by  Soleil  in  Paris.  In  the 
year  1868  appeared  the  first  model  of  a  surface  of  the 


278  HISTORY  OF  MATHEMATICS. 

third  order  with  its  twenty-seven  straight  lines,  by 
Chr.  Wiener.  In  the  sixties,  Kummer  constructed 
models  of  surfaces  of  the  fourth  order  and  of  certain 
focal  surfaces.  His  pupil  Schwarz  likewise  constructed 
a  series  of  models,  among  them  minimal  surfaces  and 
the  surfaces  of  centers  of  the  ellipsoid.  At  a  meeting 
of  mathematicians  in  Gottingen  there  was  made  a 
notable  exhibition  of  models  which  stimulated  further 
work  in  this  direction. 

In  wider  circles  the  works  suggested  by  A.  Brill, 
F.  Klein,  and  W.  Dyck  in  the  mathematical  seminar 
of  the  Munich  polytechnic  school  have  found  recogni- 
tion. There  appeared  from  1877  to  1890  over  a  hun- 
dred models  of  the  most  various  kinds,  of  value  not 
only  in  mathematical  teaching  but  also  in  lectures  on 
perspective,  mechanics  and  mathematical  physics. 

In  other  directions  also  has  illustrative  material  of 
this  sort  been  multiplied,  such  as  surfaces  of  the  third 
order  by  Rodenberg,  thread  models  of  surfaces  and 
gauche  curves  of  the  fourth  order  by  Rohn,  H.Wiener, 

and  others. 

* 
*  * 

If  one  considers  geometric  science  as  a  whole,  it 
cannot  be  denied  that  in  its  field  no  essential  differ- 
ence between  modern  analytic  and  modern  synthetic 
geometry  any  longer  exists.  The  subject  matter  and 
the  methods  of  proof  in  both  directions  have  gradu- 
ally taken  almost  the  same  form.  Not  only  does  the 
synthetic  method  make  use  of  space  intuition;  the 


GEOMETRY.  279 

analytic  representations  also  are  nothing  less  than  a 
clear  expression  of  space  relations.  And  since  metric 
properties  of  figures  may  be  regarded  as  relations  of 
the  same  to  a  fundamental  form  of  the  second  order, 
to  the  great  circle  at  infinity,  and  thus  can  be  brought 
into  the  aggregate  of  projective  properties,  instead  of 
analytic  and  synthetic  geometry,  we  have  only  a  pro- 
jective geometry  which  takes  the  first  place  in  the 
science  of  space.* 

The  last  decades,  especially  of  the  development  of 
German  mathematics,  have  secured  for  the  science  a 
leading  place.  In  general  two  groups  of  allied  works 
may  be  recognized,  f  In  the  treatises  of  the  one  ten- 
dency "after  the  fashion  of  a  Gauss  or  a  Dirichlet, 
the  inquiry  is  concentrated  upon  the  exactest  possible 
limitation  of  the  fundamental  notions"  in  the  theory 
of  functions,  theory  of  numbers,  and  mathematical 
physics.  The  investigations  of  the  other  tendency, 
as  is  to  be  seen  in  Jacobi  and  Clebsch,  start  "from  a 
small  circle  of  already  recognized  fundamental  con- 
cepts and  aim  at  the  relations  and  consequences  which 
spring  from  them,"  so  as  to  serve  modern  algebra  and 
geometry. 

On  the  whole,  then,  we  may  say  that|  "mathe- 
matics have  steadily  advanced  from  the  time  of  the 
Greek  geometers.  The  achievements  of  Euclid,  Archi- 
medes, and  Apollonius  are  as  admirable  now  as  they 

*  F.Klein,  t  Clebsch. 

tCayley,  A.,  Address  to  the  British  Association,  etc.,  1883. 


280  HISTORY  OF  MATHEMATICS. 

were  in  their  own  days.  Descartes's  method  of  co- 
ordinates is  a  possession  forever.  But  mathematics 
have  never  been  cultivated  more  zealously  and  dili- 
gently, or  with  greater  success,  than  in  this  century — 
in  the  last  half  of  it,  or  at  the  present  time :  the  ad- 
vances made  have  been  enormous,  the  actual  field  is 
boundless,  the  future  is  full  of  hope." 


V.  TRIGONOMETRY. 

A.    GENERAL  SURVEY. 

^TRIGONOMETRY  was  developed  by  the  ancients 
-*-  for  purposes  of  astronomy.  In  the  first  period  a 
number  of  fundamental  formulae  of  trigonometry  were 
established,  though  not  in  modern  form,  by  the  Greeks 
and  Arabs,  and  employed  in  calculations.  The  second 
period,  which  extends  from  the  time  of  the  gradual 
rise  of  mathematical  sciences  in  the  earliest  Middle 
Ages  to  the  middle  of  the  seventeenth  century,  estab- 
lishes the  science  of  calculation  with  angular  func- 
tions and  produces  tables  in  which  the  sexagesimal 
division  is  replaced  by  decimal  fractions,  which  marks 
a  great  advance  for  the  purely  numerical  calculation. 
During  the  third  period,  plane  and  spherical  trigo- 
nometry develop,  especially  polygonometry  and  poly- 
hedrometry  which  are  almost  wholly  new  additions  to 
the  general  whole.  Further  additions  are  the  projec- 
tive  formulae  which  have  furnished  a  series  of  inter- 
esting results  in  the  closest  relation  to  projective  ge- 
ometry. 


282  HISTORY  OF  MATHEMATICS. 

B.    FIRST  PERIOD. 

FROM  THE  MOST  ANCIENT  TIMES  TO  THE  ARABS. 

The  Papyrus  of  Ahmes*  speaks  of  a  quotient 
called  seqt.  After  observing  that  the  great  p3'ramids 
all  possess  approximately  equal  angles  of  inclination, 
the  assumption  is  rendered  probable  that  this  seqt  is 
identical  with  the  cosine  of  the  angle  which  the  edge 
of  the  pyramid  forms  with  the  diagonal  of  the  square 
base.  This  angle  is  usually  52°.  In  the  Egyptian 
monuments  which  have  steeper  sides,  the  seqt  ap- 
pears to  be  equal  to  the  trigonometric  tangent  of  the 
angle  of  inclination  of  one  of  the  faces  to  the  base. 

Trigonometric  investigations  proper  appear  first 
among  the  Greeks.  Hypsicles  gives  the  division  of 
the  circumference  into  three  hundred  sixty  degrees, 
which,  indeed,  is  of  Babylonian  origin  but  was  first 
turned  to  advantage  by  the  Greeks.  After  the  intro- 
duction of  this  division  of  the  circle,  sexagesimal 
fractions  were  to  be  found  in  all  the  astronomical  cal- 
culations of  antiquity  (with  the  single  exception  of 
Heron),  till  finally  Peurbach  and  Regiomontanus  pre- 
pared the  way  for  the  decimal  reckoning.  Hipparchus 
was  the  first  to  complete  a  table  of  chords,  but  of  this 
we  have  left  only  the  knowledge  of  its  former  exist- 

*  Cantor,  I.,  p.  58. 


TRIGONOMETRY.  283 

ence.  In  Heron  are  found  actual  trigonometric  for- 
mulae with  numerical  ratios  for  the  calculation  of  the 
areas  of  regular  polygons  and  in  fact  all  the  values  of 

cotf  —  jfor  «  =  3,  4,  ...  11,  12  are  actually  computed.  * 
Menelaus  wrote  six  books  on  the  calculation  of  chords, 
but  these,  like  the  tables  of  Hipparchus,  are  lost.  On 
the  contrary,  three  books  of  the  Spherics  of  Menelaus 
are  known  in  Arabic  and  Hebrew  translations.  These 
contain  theorems  on  transversals  and  on  the  congru- 
ence of  spherical  as  well  as  plane  triangles,  and  for 
the  spherical  triangle  the  theorem  that  a  -\-  b  -\-  c  <  ±R, 


The  most  important  work  of  Ptolemy  consists  in 
the  introduction  of  a  formal  spherical  trigonometry 
for  astronomical  purposes.  The  thirteen  books  of  the 
Great  Collection  which  contain  the  Ptolemaic  astron- 
omy and  trigonometry  were  translated  into  Arabic, 
then  into  Latin,  and  in  the  latter  by  a  blending  of  the 
Arabic  article  al  with  a  Greek  word  arose  the  word 
Almagest,  now  generally  applied  to  the  great  work  of 
Ptolemy.  Like  Hypsicles,  Ptolemy  also,  after  the 
ancient  Babylonian  fashion,  divides  the  circumfer- 
ence into  three  hundred  sixty  degrees,  but  he,  in  ad- 
dition to  this,  bisects  every  degree.  As  something 
new  we  find  in  Ptolemy  the  division  of  the  diameter 
of  the  circle  into  one  hundred  twenty  equal  parts, 
from  which  were  formed  after  the  sexagesimal  fashion 

*  Tannery  in  Mlm.  Bord.,  1881. 


284  HISTORY  OF  MATHEMATICS. 

two  classes  of  subdivisions.  In  the  later  Latin  trans- 
lations these  sixtieths  of  the  first  and  second  kind 
were  called  respectively  paries  minutae  primae  and 
paries  minutae  secundae.  Hence  came  the  later  terms 
"minutes"  and  "seconds."  Starting  from  his  theo- 
rem upon  the  inscribed  quadrilateral,  Ptolemy  calcu- 
lates the  chords  of  arcs  at  intervals  of  half  a  degree. 
But  he  develops  also  some  theorems  of  plane  and 
especially  of  spherical  trigonometry,  as  for  example 
theorems  regarding  the  right  angled  spherical  tri- 
angle. 

A  further  not  unimportant  advance  in  trigonom- 
etry is  to  be  noted  in  the  works  of  the  Hindus.  The 
division  of  the  circumference  is  the  same  as  that  of 
the  Babylonians  and  Greeks ;  but  beyond  that  there 
is  an  essential  deviation.  The  radius  is  not  divided 
sexagesimally  after  the  Greek  fashion,  but  the  arc  of 
the  same  length  as  the  radius  is  expressed  in  min- 
utes ;  thus  for  the  Hindus  r  =  3438  minutes.  Instead 
of  the  whole  chords  (^jiva),  the  half  chords  (ardhajya^ 
are  put  into  relation  with  the  arc.  In  this  relation  of 
the  half-chord  to  the  arc  we  must  recognize  the  most 
important  advance  of  trigonometry  among  the  Hindus. 
In  accordance  with  this  notion  they  were  therefore 
familiar  with  what  we  now  call  the  sine  of  an  angle. 
Besides  this  they  calculated  the  ratios  corresponding 
to  the  versed  sine  and  the  cosine  and  gave  them  spe- 
cial names,  calling  the  versed  sine  utkramajya,  the 
cosine  kotijya.  They  also  knew  the  formula  sin2*/ 


TRIGONOMETRY.  285 

-\-cos2a  =  l.  They  did  not,  however,  apply  their 
trigonometric  knowledge  to  the  solution  of  plane  tri- 
angles, but  with  them  trigonometry  was  inseparably 
connected  with  astronomical  calculations. 

As  in  the  rest  of  mathematical  science,  so  in  trig- 
onometry, were  the  Arabs  pupils  of  the  Hindus,  and 
still  more  of  the  Greeks,  but  not  without  important 
devices  of  their  own.  To  Al  Battani  it  was  well  known 
that  the  introduction  of  half  chords  instead  of  whole 
chords,  as  these  latter  appear  in  the  Almagest,  and 
therefore  reckoning  with  the  sine  of  an  angle,  is  of 
essential  advantage  in  the  applications.  In  addition 
to  the  formulae  found  in  the  Almagest,  Al  Battani 
gives  the  relation,  true  for  the  spherical  triangle, 

In  the  considera- 


tion of  right-angled  triangles  in  connection  with 
shadow-measuring,  we  find  the  quotients  -  —  and 

These  were  reckoned  for  each  degreee  by  Al 
sin  a 
Battani  and  arranged  in  a  small  table.     Here  we  find 

the  beginnings  of  calculation  with  tangents  and  co- 
tangents. These  names,  however,  were  introduced 
much  later.  The  origin  of  the  term  "sine"  is  due  to 
Al  Battani.  His  work  upon  the  motion  of  the  stars* 
was  translated  into  Latin  by  Plato  of  Tivoli,  and  this 
translation  contains  the  word  sinus  for  half  chord. 
In  Hindu  the  half  chord  was  called  ardhajya  or  also 
jiva  (which  was  used  originally  only  for  the  whole 

*  Cantor,  I.,  p.  693,  where  this  account  is  considered  somewhat  doubtful. 


286  HISTORY  OF  MATHEMATICS. 

chord);  the  latter  word  the  Arabs  adopted,  simply 
by  reason  of  its  sound,  as  jiba.  The  consonants  of 
this  word,  which  in  Arabic  has  no  meaning  of  its 
own,  might  be  read  jaib  =  bosom,  or  incision,  and 
this  pronunciation,  which  apparently  was  naturalized 
comparatively  soon  by  the  Arabs,  Plato  of  Tivoli 
translated  properly  enough  into  sinus.  Thus  was  in- 
troduced the  first  of  the  modern  names  of  the  trigo- 
nometric functions. 

Of  astronomical  tables  there  was  no  lack  at  that 

time.    Abul  Wafa,  by  whom  the  ratio was  called 

cos  a 

the  "shadow"  belonging  to  the  angle  a,  calculated  a 
table  of  sines  at  intervals  of  half  a  degree  and  also  a 
table  of  tangents,  which  however  was  used  only  for 
determining  the  altitude  of  the  sun.  About  the  same 
time  appeared  the  hakimitic  table  of  sines  which  Ibn 
Yunus  of  Cairo  was  required  to  construct  by  direction 
of  the  Egyptian  ruler  Al  Hakim.* 

Among  the  Western  Arabs  the  celebrated  astron- 
omer Jabir  ibn  Aflah,  or  Geber,  wrote  a  complete  trigo- 
nometry (principally  spherical)  after  a  method  of  his 
own,  and  this  work,  rigorous  throughout  in  its  proofs, 
was  published  in  the  Latin  edition  of  his  Astronomy 
by  Gerhard  of  Cremona.  This  work  contains  a  col- 
lection of  formulae  upon  the  right-angled  spherical 
triangle.  In  the  plane  trigonometry  he  does  not  go 

*  Cantor,  I.,  p.  743. 


TRIGONOMETRY.  287 

beyond  the  Almagest,  and  hence  he  here  deals  only 
with  whole  chords,  just  as  Ptolemy  had  taught. 


C.    SECOND  PERIOD. 

FROM  THE  MIDDLE  AGES  TO  THE  MIDDLE  OF  THE  SEVEN- 
TEENTH CENTURY. 

Of  the  mathematicians  outside  of  Germany  in  this 
period,  Vieta  made  a  most  important  advance  by  his 
introduction  of  the  reciprocal  triangle  of  a  spherical 
triangle.  In  Germany  the  science  was  advanced  by 
Regiomontanus  and  in  its  elements  was  presented  with 
such  skill  and  thorough  knowledge  that  the  plan  laid 
out  by  him  has  remained  in  great  part  up  to  the  pres- 
ent day.  Peurbach  had  already  formed  the  plan  of 
writing  a  trigonometry  but  was  prevented  by  death. 
Regiomontanus  was  able  to  carry  out  Peurbach's  idea 
by  writing  a  complete  plane  and  spherical  trigonom- 
etry. After  a  brief  geometric  introduction  Regiomon- 
tanus's  trigonometry  begins  with  the  right-angled  tri- 
angle, the  formulae  needed  for  its  computation  being 
derived  in  terms  of  the  sine  alone  and  illustrated  by 
numerical  examples.  The  theorems  on  the  right- 
angled  triangle  are  used  for  the  calculation  of  the 
equilateral  and  isosceles  triangles.  Then  follow  the 
principal  cases  of  the  oblique  angled  triangle  of  which 
the  first  (a  from  a,  b,  c)  is  treated  with  much  detail. 
The  second  book  contains  the  sine  theorem  and  a 


288  HISTORY  OF  MATHEMATICS. 

series  of  problems  relating  to  triangles.  The  third, 
fourth,  and  fifth  books  bring  in  spherical  trigonometry 
with  many  resemblances  to  Menelaus ;  in  particular 
the  angles  are  found  from  the  sides.  The  case  of  the 
plane  triangle  (a  from  a,  b,  c~}t  treated  with  consider 
able  prolixity  by  Regiomontanus,  received  a  shorter 
treatment  from  Rhaeticus,  who  established  the  for- 
mula cot^a  = ,  where  p  is  the  radius  of  the  in- 
scribed circle. 

In  this  period  were  also  published  Napier's  equa- 
tions, or  analogies.  They  express  a  relation  between 
the  sum  or  difference  of  two  sides  (angles)  and  the 
third  side  (angle)  and  the  sum  or  difference  of  the  two 
opposite  angles  (sides). 

Of  modern  terms,  as  already  stated,  the  word 
"sine"  is  the  oldest.  About  the  end  of  the  sixteenth 
century,  or  the  beginning  of  the  seventeenth,  the  ab- 
breviation cosine  for  complementi  sinus  was  introduced 
by  the  Knglishman  Gunter  (died  1626).  The  terms 
tangent  and  secant  were  first  used  by  Thomas  Finck 
(1583);  the  term  versed  sine  was  used  still  earlier.* 

By  some  writers  of  the  sixteenth  century,  e.  g.,  by 
Apian,  sinus  rectus  secundus  was  written  instead  of  co- 
sine. Rhaeticus  and  Vieta  have  perpendiculum  and 
basis  for  sine  and  cosine. f  The  natural  values  of  the 
cosine,  whose  logarithms  were  called  by  Kepler  "anti- 

*Baltzer,  R.,  Die  Elemente  der  Mathematik,  1885. 
tPfleiderer,  Trigonometrie,  1802. 


TRIGONOMETRY.  289 

logarithms,"  are  first  found  calculated  in  the  trigo- 
nometry of  Copernicus  as  published  by  Rhaeticus.* 

The  increasing  skill  in  practical  computation,  and 
the  need  of  more  accurate  values  for  astronomical 
purposes,  led  in  the  sixteenth  century  to  a  strife  after 
the  most  complete  trigonometric  tables  possible.  The 
preparation  of  these  tables,  inasmuch  as  the  calcula- 
tions were  made  without  logarithms,  was  very  tedious. 
Rhaeticus  alone  had  to  employ  for  this  purpose  a 
number  of  computers  for  twelve  years  and  spent 
thereby  thousands  of  gulden,  f 

The  first  table  of  sines  of  German  origin  is  due  to 
Peurbach.  He  put  the  radius  equal  to  600  000  and 
computed  at  intervals  of  10'  (in  Ptolemy  r  =  6Q,  with 
some  of  the  Arabs  r  =  150).  Regiomontanus  com- 
puted two  new  tables  of  sines,  one  for  r=6  000  000, 
the  other,  of  which  no  remains  are  left,  for  r  = 
10  000  000.  Besides  these  we  have  from  Regiomon- 
tanus a  table  of  tangents  for  every  degree,  r  =  100  000. 
The  last  two  tables  evidently  show  a  transition  from 
the  sexagesimal  system  to  the  decimal.  A  table  of 
sines  for  every  minute,  with  r  =  100  000,  was  pre- 
pared by  Apian. 

In  this  field  should  also  be  mentioned  the  indefat- 
igable perseverance  of  Joachim  Rhaeticus.  He  did 
not  associate  the  trigonometric  functions  with  the 
arcs  of  circles,  but  started  with  the  right-angled  tri- 

*M.  Curtze,  in  Schlomilch' s  Zeitschrift,  Bd.  XX. 

1  Gerhardt,  Geschichte  der  Mathematik  in  Deutschland  1877. 


2go  HISTORY  OF  MATHEMATICS. 

angle  and  used  the  terms  perpendiculum  for  sine,  basis 
for  cosine.  He  calculated  (partly  himself  and  partly 
by  the  help  of  others)  the  first  table  of  secants ;  later, 
tables  of  sines,  tangents,  and  secants  for  every  10", 
for  radius  =10000  millions,  and  later  still,  for  r  — 
1016.  After  his  death  the  whole  work  was  published 
by  Valentin  Otho  in  the  year  1596  in  a  volume  of  1468 
pages.* 

To  the  calculation  of  natural  trigonometric  func 
tions  Bartholomaeus  Pitiscus  also  devoted  himself. 
Tn  the  second  book  of  his  Trigonometry  he  sets  forth 
his  views  on  computations  of  this  kind.  His  tables 
contain  values  of  the  sines,  tangents,  and  secants  on 
the  left,  and  of  the  complements  of  the  sines,  tangents 
and  secants  (for  so  he  designated  the  cosines,  cotan- 
gents, and  cosecants)  on  the  right.  There  are  added 
proportional  parts  for  1',  and  even  for  10".  In  the 
whole  calculation  r  is  assumed  equal  to  1026.  The 
work  of  Pitiscus  appeared  at  the  beginning  of  the 
seventeenth  century. 

The  tables  of  the  numerical  values  of  the  trigono- 
metric functions  had  now  attained  a  high  degree  of 
accuracy,  but  their  real  significance  and  usefulness 
were  first  shown  by  the  introduction  of  logarithms. 

Napier  is  usually  regarded  as  the  inventor  of  log- 
arithms, although  Cantor's  review  of  the  evidencef 
leaves  no  room  for  doubt  that  Biirgi  was  an  indepen- 
dent discoverer.  His  Progress  Tabulen,  computed  be- 

*  Gerhardt.  t  Cantor,  II.,  pp.  662  et  seq. 


TRIGONOMETRY.  29  I 

tween  1603  and  1611  but  not  published  until  1620  is 
really  a  table  of  antilogarithms.  Biirgi's  more  gen- 
eral point  of  view  should  also  be  mentioned.  He  de- 
sired to  simplify  all  calculations  by  means  of  loga- 
rithms while  Napier  used  only  the  logarithms  of  the 
trigonometric  functions. 

Biirgi  was  led  to  this  method  of  procedure  by 
comparison  of  the  two  series  0,  1,  2,  3,  ...  and  1,  2, 
4,  8,  ...  or  2»,  21,  22,  2»,  .  .  .  He  observed  that  for 
purposes  of  calculation  it  was  most  convenient  to  se- 
lect 10  as  the  base  of  the  second  series,  and  from  this 
standpoint  he  computed  the  logarithms  of  ordinary 
numbers,  though  he  first  decided  on  publication  when 
Napier's  renown  began  to  spread  in  Germany  by  rea- 
son of  Kepler's  favorable  reports.  Biirgi's  Geometri- 
sche  Progress  Tabulen  appeared  at  Prague  in  1620,* 
and  contained  the  logarithms  of  numbers  from  108  to 
109  by  tens.  Burgi  did  not  use  the  term  logarithmus, 
but  by  reason  of  the  way  in  which  they  were  printed 
he  called  the  logarithms  "red  numbers,"  the  numbers 
corresponding,  "black  numbers." 

Napier  started  with  the  observation  that  if  in  a 
circle  with  two  perpendicular  radii  OA§  and  OA\ 
(r  =  l),  the  sine  S0Si  ||  OA±  moves  from  O  to  AQ  at 
intervals  forming  an  arithmetic  progression,  its  value 
decreases  in  geometrical  progression.  The  segment 
OS0,  Napier  originally  called  numerus  artificialis  and 
later  the  direction  number  or  logarithmus.  The  first 

*Gerhardt. 


2Q2  HISTORY  OF  MATHEMATICS. 

publication  of  this  new  method  of  calculation,  in  which 
r=107,  log  sin  60°  =  0,  log  sin  0°  =  oo,  so  that  the  log- 
arithms increased  as  the  sines  decreased,  appeared  in 
1614  and  produced  a  great  sensation.  Henry  Briggs 
had  studied  Napier's  work  thoroughly  and  made  the 
important  observation  that  it  would  be  more  suitable 
for  computation  if  the  logarithms  were  allowed  to  in- 
crease with  the  numbers.  He  proposed  to  put  log  1 
=  0,  log  10  =  1,  and  Napier  gave  his  assent.  The  ta- 
bles of  logarithms  calculated  by  Briggs,  on  the  basis 
of  this  proposed  change,  for  the  natural  numbers  from 
1  to  20  000  and  from  90  000  to  100  000  were  reckoned 
to  14  decimal  places.  The  remaining  gap  was  filled  by 
the  Dutch  bookseller  Adrian  Vlacq.  His  tables  which 
appeared  in  the  year  1628  contained  the  logarithms  of 
numbers  from  1  to  100  000  to  10  decimal  places.  In 
these  tables,  under  the  name  of  his  friend  De  Decker, 
Vlacq  introduced  logarithms  upon  the  continent.  As- 
sisted by  Vlacq  and  Gellibrand,  Briggs  computed  a 
table  of  sines  to  fourteen  places  and  a  table  of  tan- 
gents and  secants  to  ten  places,  at  intervals  of  36". 
These  tables  appeared  in  1633.  Towards  the  close  of 
the  seventeenth  century  Claas  Vooght  published  a 
table  of  sines,  tangents,  and  secants  with  their  loga- 
rithms, and,  what  was  especially  remarkable,  they 
were  engraved  on  copper. 

Thus  was  produced  a  collection  of  tables  for  logarithmic  com- 
putation valuable  for  all  time.  This  was  extended  by  the  intro- 
duction of  the  addition  and  subtraction  logarithms  always  named 


TRIGONOMETRY.  293 

after  Gauss,  but  whose  inventor,  according  to  Gauss's  own  testi 
mony,  is  Leonelli.  The  latter  had  proposed  calculating  a  table 
with  fourteen  decimals  ;  Gauss  thought  this  impracticable,  and 
calculated  for  his  own  use  a  table  with  five  decimals.* 

In  the  year  1875  there  were  in  existence  553  logarithmic  tables 
with  decimal  places  ranging  in  number  from  3  to  102.  Arranged 
according  to  frequency,  the  7 -place  tables  stand  at  the  head,  then 
follow  those  with  5  places,  6-places,  4-places,  and  10-places.  The 
only  table  with  102  places  is  found  in  a  work  by  H.  M.  Parkhurst 
(Astronomical  Tables,  New  York,  1871). 

Investigations  of  the  errors  occurring  in  logarithmic  tables 
have  been  made  by  J.  W.  L.  Glaisher.  f  It  was  there  shown  that 
every  complete  table  had  been  transcribed,  directly  or  indirectly 
after  a  more  or  less  careful  revision,  from  the  table  published  in 
1628  which  contains  the  results  of  Briggs's  Arithmetica  logarith- 
mica  of  1624  for  numbers  from  1  to  100000  to  ten  places.  In  the 
first  seven  places  Glaisher  found  171  errors  of  which  48  occur  in 
the  interval  from  1  to  10000.  These  errors,  due  to  Vlacq,  have 
gradually  disappeared.  Of  the  mistakes  in  Vlacq,  98  still  appear 
in  Newton  (1658),  19  in  Gardiner  (1742),  5  in  Vega  (1797),  2  in 
Callet  (1855),  2  in  Sang  (1871).  Of  the  tables  tested  by  Glaisher, 
four  turned  out  to  be  free  from  error,  viz.,  those  of  Bremiker 
(1857),  Schron  (I860),  Callet  (1862),  and  Bruhns  (1870).  Contribu- 
tions to  the  rapid  calculation  of  common  logarithms  have  been 
made  by  Koralek  (1851)  and  R.  Hoppe  (1876) ;  the  work  of  the 
latter  is  based  upon  the  theorem  that  every  positive  number  may 
be  transformed  into  an  infinite  product.:): 

*  Gauss,  Werke,  III.,  p.  244.     Porro  in  Bone.  Bull.,  XVIII. 

t  Fortschritte,  1873. 

$  Stolz,  Vorlesungen  uber  allgemeine  Arithmetik,  1885-1886. 


294  HISTORY  OF  MATHEMATICS. 


D.    THIRD  PERIOD. 

FROM  THE  MIDDLE  OF  THE  SEVENTEENTH  CENTURY 
TO  THE  PRESENT. 

After  Regiomontanus  had  laid  the  foundations  of 
plane  and  spherical  trigonometry,  and  his  successors 
had  made  easier  the  work  of  computation  by  the  com- 
putation of  the  numerical  values  of  the  trigonomet- 
ric functions  and  the  creation  of  a  serviceable  sys- 
tem of  logarithms,  the  inner  structure  of  the  science 
was  ready  to  be  improved  in  details  during  this  third 
period.  Important  innovations  are  especially  due  to 
EuLer,  who  derived  the  whole  of  spherical  trigonom- 
etry from  a  few  simple  theorems.  Euler  denned  the 
trigonometric  functions  as  mere  numbers,  so  as  to  be 
able  to  substitute  them  for  series  in  whose  terms  ap- 
pear arcs  of  circles  from  which  the  trigonometric  func- 
tions proceed  according  to  definite  laws.  From  him 
we  have  a  number  of  trigonometric  formulae,  in  part 
entirely  new,  and  in  part  perfected  in  expression. 
These  were  made  especially  clear  when  Euler  denoted 
the  elements  of  the  triangle  by  a,  b,  c,  a,  ft,  y.  Then 
such  expressions  as  sin  a,  tana  could  be  introduced 
where  formerly  special  letters  had  been  used  for  the 
same  purpose.*  Lagrange  and  Gauss  restricted  them- 
selves to  a  single  theorem  in  the  derivation  of  spheri- 
cal trigonometry.  The  system  of  equations 

*Baltzer,  R.,  Die  Elemente  der  Mathematik,  1885. 


TRIGONOMETRY. 


295 


a     .    b  -4-  c 


sm  -2 

with  the  corresponding  relations,  is  ordinarily  ascribed 
to  Gauss,  though  the  equations  were  first  published 
by  Delambre  in  1807  (by  Mollweide  1808,  by  Gauss 
1809).*  The  case  of  the  Pothenot  problem  is  similar: 
it  was  discussed  by  Snellius  1614,  by  Pothenot  1692, 
by  Lambert  1765.f 

The  principal  theorems  of  polygonometry  and 
polyhedrometry  were  established  in  the  eighteenth 
century.  To  Euler  we  owe  the  theorem  on  the  area 
of  the  orthogonal  projection  of  a  plane  figure  upon 
another  plane  ;  to  Lexell  the  theorem  upon  the  pro- 
jection of  a  polygonal  line.  Lagrange,  Legendre, 
Carnot  and  others  stated  trigonometric  theorems  for 
polyhedra  (especially  the  tetrahedra),  Gauss  for  the 
spherical  quadrilateral. 

The  nineteenth  century  has  given  to  trigonometry 
a  series  of  new  formulae,  the  so-called  projective  for- 
mulae. Besides  Poncelet,  Steiner,  and  Gudermann, 
Mobius  deserves  special  mention  for  having  devised 
a  generalization  of  spherical  trigonometry,  such  that 
sides  or  angles  of  a  triangle  may  exceed  180°.  The  im- 
portant improvements  which  in  modern  times  trigono- 
metric developments  have  contributed  to  other  mathe- 
matical sciences,  may  be  indicated  in  this  one  sentence: 
their  extended  description  would  considerably  en- 
croach upon  the  province  of  other  branches  of  science. 

*  Hammer,  Lehrbuck  der  ebetten  vnd  sphdrischen  TrigoMometrie,  1897. 
t  Baltzer,  R.,  Die  Elemente  der  Mathematik,  1885. 


BIOGRAPHICAL  NOTES.* 

Abel,  Niels  Henrik.  Born  at  Findoe,  Norway,  August  5,  1802  ; 
.  died  April  6,  1829.  Studied  in  Christiania,  and  for  a  short 
time  in  Berlin  and  Paris.  Proved  the  impossibility  of  the 
algebraic  solution  of  the  quintic  equation  ;  elaborated  the  the- 
ory of  elliptic  functions ;  founded  the  theory  of  Abelian  func- 
tions. 

Abul  Jud,  Mohammed  ibn  al  Lait  al  Shanni.  Lived  about  1050. 
Devoted  much  attention  to  geometric  problems  not  soluble 
with  compasses  and  straight  edge  alone. 

Abul  Wafa  al  Buzjani.  Born  at  Buzjan,  Persia,  June  10,  940; 
died  at  Bagdad,  July  i,  998.  Arab  astronomer.  Translated 
works  of  several  Greek  mathematicians ;  improved  trigonom- 
etry and  computed  some  tables ;  interested  in  geometric  con- 
structions requiring  a  single  opening  of  the  compasses. 

Adelard.  About  1120.  English  monk  who  journeyed  through  Asia 
Minor,  Spain,  Egypt,  and  Arabia.  Made  the  first  translation 
of  Euclid  from  Arabic  into  Latin.  Translated  part  of  Al 
Khowarazmi's  works. 

Al  Battani  (Albategnius).  Mohammed  ibn  Jabir  ibn  Sinan  Abu 
Abdallah  al  Battani.  Born  in  Battan,  Mesopotamia,  c.  850; 
died  in  Damascus,  929.  Arab  prince,  governor  of  Syria ;  great- 

*The  translators  feel  that  these  notes  will  be  of  greater  value  to  the 
reader  by  being  arranged  alphabetically  than,  as  in  the  original,  by  periods, 
especially  as  this  latter  arrangement  is  already  given  in  the  body  of  the 
work.  They  also  feel  that  they  will  make  the  book  more  serviceable  by 
changing  the  notes  as  set  forth  in  the  original,  occasionally  eliminating  mat- 
ter of  little  consequence,  and  frequently  adding  to  the  meagre  information 
given.  They  have,  for  this  purpose,  freely  used  such  standard  works  as  Can- 
tor, Hankel,  Giinther,  Zeuthen,  et  al.,  and  especially  the  valuable  little  Zeit- 
tafeln  zur  Geschichte  der  Mathematik,  Physik  und  Astronomic  bis  zum  Jahre 
1500,  by  Felix  Miiller,  Leipzig,  1893.  Dates  are  A.  D.,  except  when  prefixed 
by  the  negative  sign. 


298  HISTORY  OF  MATHEMATICS. 

est  Arab  astronomer  and  mathematician.  Improved  trigonom- 
etry and  computed  the  first  table  of  cotangents. 

Alberti,  Leo  Battista.     1404-1472.    Architect,  painter,  sculptor. 

Albertus  Magnus.  Count  Albrecht  von  Bollstadt.  Born  at  Lau- 
ingen  in  Bavaria,  1193  or  1205 ;  died  at  Cologne,  Nov.  15, 
1280.  Celebrated  theologian,  chemist,  physicist,  and  mathe- 
matician. 

.11  Biruni,  Abul  Rihan  Mohammed  ibn  Ahmed.  From  Birun, 
valley  of  the  Indus ;  died  1038.  Arab,  but  lived  and  travelled 
in  India  and  wrote  on  Hindu  mathematics.  Promoted  spheri- 
cal trigonometry. 

Alcuin.  Born  at  York,  736;  died  at  Hersfeld,  Hesse,  May  19 
804.  At  first  a  teacher  in  the  cloister  school  at  York ;  then 
assisted  Charlemagne  in  his  efforts  to  establish  schools  in 
France. 

Alhazen,  Ibn  al  Haitam.  Born  at  Bassora,  950 ;  died  at  Cairo 
1038.  The  most  important  Arab  writer  on  optics. 

Al  Kalsadi,  Abul  Hasan  AH  ibn  Mohammed.  Died  1486  or  1477. 
From  Andalusia  or  Granada.  Arithmetician. 

Al  fCarkhi,  Abu  Bekr  Mohammed  ibn  al  Hosain.  Lived  about 
1010.  Arab  mathematician  at  Bagdad.  Wrote  on  arithmetic, 
algebra  and  geometry. 

Al  Khojandi,  Abu  Mohammed.  From  Khojand,  in  Khorassan  ; 
was  living  in  992.  Arab  astronomer. 

Al  Khorvarazmi,  Abu  Jafar  Mohammed  ibn  Musa.  First  part  of 
ninth  century.  Native  of  Khwarazm  (Khiva).  Arab  mathe- 
matician and  astronomer.  The  title  of  his  work  gave  the  name 
to  algebra.  Translated  certain  Greek  works. 

Al  Kindi,  Jacob  ibn  Ishak,  Abu  Yusuf .  Born  c.  813;  died  873. 
Arab  philosopher,  physician,  astronomer  and  astrologer. 

Al  Kuhi,  Vaijan  ibn  Rustam  Abu  Sahl.  Lived  about  975.  Arab 
astronomer  and  geometrician  at  Bagdad. 

Al  Nasauui,  Abul  Hasan  Ali  ibn  Ahmed.  Lived  about  1000 
From  Nasa  in  Khorassan.  Arithmetician. 

Al  Sag-ant.  Ahmed  ibn  Mohammed  al  Sagani  Abu  Hamid  al  Us- 
turlabi.  From  Sagan,  Khorassan ;  died  990.  Bagdad  astron- 
omer 


BIOGRAPHICAL  NOTES.  2QQ 

Anaxagoras.  Born  at  Clazomene,  Ionia,  —499;  died  at  Lamp- 
sacus,  — 428.  Last  and  most  famous  philosopher  of  the  Ionian 
school.  Taught  at  Athens.  Teacher  of  Euripides  and  Pe- 
ricles. 

Apianus  (Apian),  Petrus.  Born  at  Leisnig,  Saxony,  1495  ;  died  in 
1552.  Wrote  on  arithmetic  and  trigonometry. 

Afollonius  of  Perga,  in  Pamphylia.  Taught  at  Alexandria  be- 
tween — 250  and  — 200,  in  the  reign  of  Ptolemy  Philopator. 
His  eight  books  on  conies  gave  him  the  name  of  "the  great 
geometer."  Wrote  numerous  other  works.  Solved  the  gene- 
ral quadratic  with  the  help  of  conies. 

Arbogast,  Louis  Fran§ois  Antoine.  Born  at  Mutzig,  1759 ;  died 
1803.  Writer  on  calculus  of  derivations,  series,  gamma  func- 
tion, differential  equations. 

Archimedes.  Born  at  Syracuse,  — 287(7) ;  killed  there  by  Roman 
soldiers  in  — 212.  Engineer,  architect,  geometer,  physicist. 
Spent  some  time  in  Spain  and  Egypt.  Friend  of  King  Hiero. 
Greatly  developed  the  knowledge  of  mensuration  of  geometric 
solids  and  of  certain  curvilinear  areas.  In  physics  he  is  known 
for  his  work  in  center  of  gravity,  levers,  pulley  and  screw, 
specific  gravity,  etc. 

Archytas.  Born  at  Tarentum — 430;  died — 365.  Friend  of  Plato, 
a  Pythagorean,  a  statesman  and  a  general.  Wrote  on  propor- 
tion, rational  and  irrational  numbers,  tore  surfaces  and  sec- 
tions, and  mechanics. 

Argand,  Jean  Robert.  Born  at  Geneva,  1768  ;  died  c.  1825.  Pri- 
vate life  unknown.  One  of  the  inventors  of  the  present  method 
of  geometrically  representing  complex  numbers  (1806). 

Aristotle.  Born  at  Stageira,  Macedonia,  —384 ;  died  at  Chalcis, 
Euboea,  — 322.  Founder  of  the  peripatetic  school  of  philoso- 
phy ;  teacher  of  Alexander  the  Great.  Represented  unknown 
quantities  by  letters ;  distinguished  between  geometry  and 
geodesy ;  wrote  on  physics ;  suggested  the  theory  of  combina- 
tions. 

Arydbhatta.  Born  at  Pataliputra  on  the  Upper  Ganges,  476. 
Hindu  mathematician.  Wrote  chiefly  on  algebra,  including 
quadratic  equations,  permutations,  indeterminate  equations, 
and  magic  squares. 


300  HISTORY  OF  MATHEMATICS. 

August,  Ernst  Ferdinand.  Born  at  Prenzlau,  1795  ;  died  1870  as 
director  of  the  Kolnisch  Realgymnasium  in  Berlin. 

Autolykus  of  Pitane,  Asia  Minor.  Lived  about  — 330.  Greek 
astronomer  ;  author  of  the  oldest  work  on  spherics. 

Avicenna.  Abu  AH  Hosain  ibn  Sina.  Born  at  Charmatin,  near 
Bokhara,  978  ;  died  at  Hamadam,  in  Persia,  1036.  Arab  phy- 
sician and  naturalist.  Edited  several  mathematical  and  phys- 
ical works  of  Aristotle,  Euclid,  etc.  Wrote  on  arithmetic  and 
geometry. 

Babbage,  Charles.  Born  at  Totnes,  Dec.  26,  1792 ;  died  at  Lon- 
don, Oct.  18,  1871.  Lucasian  professor  of  mathematics  at 
Cambridge.  Popularly  known  for  his  calculating  machine. 
Did  much  to  raise  the  standard  of  mathematics  in  England. 

Bachet.     See  Me*ziriac. 

Bacon,  Roger.  Born  at  Ilchester,  Somersetshire,  1214;  died  at 
Oxford,  June  u,  1294.  Studied  at  Oxford  and  Paris;  profes- 
sor at  Oxford  ;  mathematician  and  physicist. 

Balbus.     Lived  about  100.     Roman  surveyor. 

Baldi,  Bernardino.  Born  at  Urbino,  1553;  died  there,  1617. 
Mathematician  and  general  scholar.  Contributed  to  the  his- 
tory of  mathematics. 

Baltzer,  Heinrich  Richard.  Born  at  Meissen  in  1818;  died  at 
Giessen  in  1887.  Professor  of  mathematics  at  Giessen. 

Barlaam,  Bernard.     Beginning  of  fourteenth  century.     A  mcnk 

who  wrote  on  astronomy  and  geometry. 
Barozzi,  Francesco.     Italian  mathematician.     1537-1604. 

Barrozu,  Isaac.  Born  at  London,  1630;  died  at  Cambridge,  May 
4,  1677.  Professor  of  Greek  and  mathematics  at  Cambridge. 
Scholar,  mathematician,  scientist,  preacher.  Newton  was  his 
pupil  and  successor. 

Beda,  the  Venerable.  Born  at  Monkton,  near  Yarrow,  Northum- 
berland, in  672;  died  at  Yarrow,  May  26,  735.  Wrote  on  chro- 
nology and  arithmetic. 

Btttavitis,  Giusto.  Born  at  Bassano,  near  Padua,  Nov.  22,  1803; 
died  Nov.  6,  1880.  Known  for  his  work  in  projective  geom 
etry  and  his  method  of  equipollences. 


BIOGRAPHICAL  NOTES.  301 

Berndinus.  Lived  about  1020.  Pupil  of  Gerbert  at  Paris.  Wrote 
on  arithmetic. 

Bernoulli.     Famous  mathematical  family. 

Jacob  (often  called  James,  by  the  English),  born  at  Basel,  Dec. 
27,  1654 ;  died  there  Aug.  16,  1705.    Among  the  first  to  recog- 
nize the  value  of  the  calculus.     His  De  Arte  Conjectandi  is  a 
classic  on  probabilities.   Prominent  in  the  study  of  curves,  the 
logarithmic  spiral  being  engraved  on  his  monument  at  Basel. 
John  (Johann),  his  brother ;  born  at  Basel,  Aug.  7,  1667 ;  died 
there  Jan.  i,  1748.     Made  the  first  attempt  to  construct  an 
integral  and  an  exponential   calculus.     Also  prominent  as  a 
physicist,  but  his  abilities  were  chiefly  as  a  teacher. 
Nicholas  (Nikolaus),  his  nephew  ;  born  at  Basel,  Oct.  10,  1687 ; 
died  there  Nov.  29,  1759.    Professor  at  St.  Petersburg,  Basel, 
and  Padua.   Contributed  to  the  study  of  differential  equations. 
Daniel,  son  of  John  ;   born  at  Groningen,  Feb.  9,  1700 ;  died  at 
Basel  in  1782.     Professor  of  mathematics  at  St.  Petersburg. 
His  chief  work  was  on  hydrodynamics. 
John  the  younger,  son  of  John.    1710-1790.    Professor  at  Basel. 

Bezout,  Etienne.  Born  at  Namours  in  1730 ;  died  at  Paris  in 
1783.  Algebraist,  prominent  in  the  study  of  symmetric  func- 
tions and  determinants. 

Bhaskara  Acharya.  Born  in  1114.  Hindu  mathematician  and 
astronomer.  Author  of  the  Lilavati  and  the  Vijaganita,  con- 
taining the  elements  of  arithmetic  and  algebra.  One  of  the 
most  prominent  mathematicians  of  his  time. 

Biot,  Jean  Baptiste.  Born  at  Paris,  Apr.  21,  1774 ;  died  same 
place  Feb.  3,  1862.  Professor  of  physics,  mathematics,  as- 
tronomy. Voluminous  writer. 

Boethius,  Anicius  Manlius  Torquatus  Severinus.  Born  at  Rome, 
480  ;  executed  at  Pavia,  524.  Founder  of  medieval  scholasti 
cism.  Translated  and  revised  many  Greek  writings  on  math- 
ematics, mechanics,  and  physics.  Wrote  on  arithmetic.  While 
in  prison  he  composed  his  Consolations  of  Philosophy. 

Bolyai:   Wolfgang  Bolyai  de  Bolya.     Born  at  Bolya,  1775  ;  died 

in  1856.     Friend  of  Gauss. 

Johann  Bolyai  de  Bolya,  his  son.  Born  at  Klausenburg,  1802  ; 
died  at  Maros-Vasarhely,  1860.  One  of  the  discoverers  (see 
Lobachevsky)  of  the  so-called  non-Euclidean  geometry. 


3O2  HISTORY  OF  MATHEMATICS. 

Bolzano,    Bernhard.     1781-1848.     Contributed    to   the  study   of 

series. 
Bonibelli,   Rafaele.     Italian.     Born  c.   1530.     His  algebra  (1572) 

summarized  all  then  known  on  the  subject.     Contributed  to 

the  study  of  the  cubic. 

Boncomfagni,  Baldassare.  Wealthy  Italian  prince.  Born  at  Rome. 
May  10,  1821 ;  died  at  same  place,  April  12,  1894.  Publisher 
of  Boncompagni's  Bulletino. 

Boole,  George.  Born  at  Lincoln,  1815  ;  died  at  Cork,  1864.  Pro- 
fessor of  mathematics  in  Queen's  College,  Cork  The  theory 
of  invariants  and  covariants  may  be  said  to  start  with  his  con- 
tributions (1841). 

Booth,  James.  1806-1878.  Clergyman  and  writer  on  elliptic  in- 
tegrals. 

Borchardt,  Karl  Wilhelm.  Born  in  1817;  died  at  Berlin,  1880 
Professor  at  Berlin. 

Boschi,  Pietro.     Born  at  Rome,   1833  ;  died  in  1887.     Professor 

at  Bologna. 
Bouquet,  Jean  Claude.     Born  at  Morteau  in  1819;  died  at  Paris, 

1885. 
Bour,  Jacques  Edmond  fimile.  Born  in  1832;  died  at  Paris,  1866 

Professor  in  the  ficole  Polytechnique. 

Bradzvardine,  Thomas  de.     Born  at  Hard  field,  near  Chichester. 

1290  ;  died  at  Lambeth,  Aug.  26,  1349.    Professor  of  theolog\ 

at  Oxford  and  later  Archbishop  of  Canterbury.     Wrote  upo:: 

arithmetic  and  geometry. 
Brahmagtifta.     Born  in  598.     Hindu  mathematician.    Contrib 

uted  to  geometry  and  trigonometry. 

Brasseur,  Jean  Baptiste.     1802-1868.     Professor  at  Liege. 

Bretschneider ;  Carl  Anton.  Born  at  Schneeberg,  May  27,  1808  , 
died  at  Gotha,  November  6,  1878. 

Brianchon,  Charles  Julien.  Born  at  Sevres,  1785  ;  died  in  1864. 
Celebrated  for  his  reciprocal  (1806)  to  Pascal's  mystic  hexa- 
gram 

Briggs,  Henry.  Born  at  Warley  Wood,  near  Halifax,  Yorkshire, 
Feb.  1560-1  ;  died  at  Oxford  Jan.  26,  1630-1.  Savilian  Pro- 


BIOGRAPHICAL  NOTES.  303 

fessor  of  geometry  at  Oxford.  Among  the  first  to  recognize 
the  value  of  logarithms;  those  with  decimal  base  bear  his 
name. 

Briot,  Charles  August  Albert.     Born  at  Sainte-Hippolyte,  1817; 

died  in  1882.     Professor  at  the  Sorbonne,  Paris. 
Brouncker,  William,  Lord.  Born  in  1620  (?) ;  died  at  Westminster, 

1684.     First  president  of  the  Royal  Society.     Contributed  to 

the  theory  of  series. 

Brunetteschi,  Filippo.  Born  at  Florence,  1379;  died  there  April 
16,  1446.  Noted  Italian  architect. 

Btirgi,  Joost  (Jobst).  Born  at  Lichtensteig,  St.  Gall,  Switzerland, 
1552  ;  died  at  Cassel  in  1632.  One  of  the  first  to  suggest  a 
system  of  logarithms.  The  first  to  recognize  the  value  of  mak- 
ing the  second  member  of  an  equation  zero. 

Caporali,  Ettore.  Born  at  Perugia,  1855  ;  died  at  Naples,  1886. 
Professor  of  mathematics  and  writer  on  geometry. 

Cardan,  Jerome  (Hieronymus,  Girolamo).  Born  at  Pavia,  1501 ; 
died  at  Rome,  1576.  Professor  of  mathematics  at  Bologna  and 
Padua.  Mathematician,  physician,  astrologer.  Chief  contri- 
butions to  algebra  and  theory  of  epicycloids. 

Carnot,  Lazare  Nicolas  Marguerite.  Born  at  Nolay,  Cote  d'Or, 
1753  ;  died  in  exile  at  Magdeburg,  1823.  Contributed  to  mod- 
ern geometry. 

Cassini,  Giovanni  Domenico.  Born  at  Perinaldo,  near  Nice,  1625; 
died  at  Paris,  1712.  Professor  of  astronomy  at  Bologna,  and 
first  of  the  family  which  for  four  generations  held  the  post  of 
director  of  the  observatory  at  Paris. 

Castigliano,  Carlo  Alberto.     1847-1884.     Italian  engineer. 

Catalan,  Eugene  Charles.  Born  at  Bruges,  Belgium,  May  30, 
1814;  died  Feb.  14,  1894.  Professor  of  mathematics  at  Paris 
and  Liege. 

~ataldi,  Pietro  Antonio.  Italian  mathematician,  born  1548  ;  died 
at  Bologna,  1626.  Professor  of  mathematics  at  Florence, 
Perugia  and  Bologna.  Pioneer  in  the  use  of  continued  frac- 
tions. 

Cattaneo,  Francesco.  1811-1875.  Professor  of  physics  and  me- 
chanics in  the  University  of  Pavia. 


304  HISTORY  OF  MATHEMATICS. 

Cauchy,  Augustin  Louis.  Born  at  Paris,  1789  ;  died  at  Sceaux, 
1857.  Professor  of  mathematics  at  Paris.  One  of  the  most 
prominent  mathematicians  of  his  time.  Contributed  to  the 
theory  of  functions,  determinants,  differential  equations,  the- 
ory of  residues,  elliptic  functions,  convergent  series,  etc. 

Cavalieri,  Bonaventura.  Born  at  Milan,  1598  ;  died  at  Bologna, 
1647.  Paved  the  way  for  the  differential  calculus  by  his 
method  of  indivisibles  (1629). 

Cayley,  Arthur.  Born  at  Richmond,  Surrey,  Aug.  16,  1821 ;  died 
at  Cambridge,  Jan.  26,  1895.  Sadlerian  professor  of  mathe- 
matics, University  of  Cambridge.  Prolific  writer  on  mathe- 
matics. 

Ceva,  Giovanni.  i6^-c.  1737.  Contributed  to  the  theory  of  trans- 
versals. 

Chasles,  Michel.  Born  at  Chartres,  Nov.  15,  1793  ;  died  at  Paris, 
Dec.  12,  1880.  Contributed  extensively  to  the  theory  of  mod- 
ern geometry. 

Chelini,  Domenico.  Born  1802;  died  Nov.  16,  1878.  Italian  mathe- 
matician ;  contributed  to  analytic  geometry  and  mechanics. 

Chuguet,  Nicolas.  From  Lyons  ;  died  about  1500.  Lived  in  Paris 
and  contributed  to  algebra  and  arithmetic. 

Clairaiit,  Alexis  Claude.  Born  at  Paris,  1713  ;  died  there,  1765. 
Physicist,  astronomer,  mathematician.  Prominent  in  the  study 
of  curves. 

Clausberg,  Christlieb  von.  Born  at  Danzig,  1689  ;  died  at  Copen- 
hagen, 1751. 

Clebsch,  Rudolf  Friedrich  Alfred.  Born  January  19,  1833  ;  died 
Nov.  7,  1872.  Professor  of  mathematics  at  Carlsruhe,  Giessen 
and  Gottingen. 

Condorcet,  Marie  Jean  Antoine  Nicolas.  Born  at  Ribemont,  near 
St.  Quentin,  Aisne.  1743;  died  at  Bourg-la  Reine,  1794.  Sec- 
retary of  the  Academic  des  Sciences.  Contributed  to  the  the- 
ory of  probabilities. 

Cotes,  Roger.  Born  at  Burbage,  near  Leicester,  July  10,  1682  ; 
died  at  Cambridge,  June  5,  1716.  Professor  of  astronomy  at 
Cambridge.  His  name  attaches  to  a  number  of  theorems  in 
geometry,  algebra  and  analysis.  Newton  remarked,  ' '  If  Cotes 
had  lived  we  should  have  learnt  something." 


BIOGRAPHICAL  NOTES.  305 

Cramer,  Gabriel.  Born  at  Geneva,  1704  ;  difid  at  Bagnols,  1752. 
Added  to  the  theory  of  equations  and  revived  the  study  of  de- 
terminants (begun  by  Leibnitz).  Wrote  a  treatise  on  curves. 

Crette,  August  Leopold.  Born  at  Eichwerder  (Wriezen  a.  d.  Oder), 
1780  ;  died  in  1855.  Founder  of  the  Journal  filr  reine  und 
angeivandte  Mathematik  (1826). 

D'  Alembert,  Jean  le  Rond.  Born  at  Paris,  1717  ;  died  there,  1783. 
Physicist,  mathematician,  astronomer.  Contributed  to  the 
theory  of  equations. 

DC  Beaune,  Florimond.  1601-1652.  Commentator  on  Descartes's 
Geometry. 

DC  la  Gournerie,  Jules  Antoine  Rene"  Maillard.  Born  in  1814 ; 
died  at  Paris,  1833.  Contributed  to  descriptive  geometry. 

Del  Monte,  Guidobaldo.  1545-1607.  Wrote  on  mechanics  and 
perspective. 

Democritus.  Born  at  Abdera,  Thrace,  — 460  ;  died  c.  — 370.  Stud- 
ied in  Egypt  and  Persia.  Wrote  on  the  theory  of  numbers  and 
on  geometry.  Suggested  the  idea  of  the  infinitesimal. 

De  Moivre,  Abraham.  Born  at  Vitry,  Champagne,  1667  ;  died  at 
London,  1754.  Contributed  to  the  theory  of  complex  num- 
bers and  of  probabilities 

De  Morgan,  Augustus.  Born  at  Madura,  Madras,  June  1806  ; 
died  March  18,  1871.  First  professor  of  mathematics  in  Uni- 
versity of  London  (1828).  Celebrated  teacher,  but  also  con- 
tributed to  algebra  and  the  theory  of  probabilities. 

Dcsargues,  Gerard.  Born  at  Lyons,  1593  ;  died  in  1662.  One  of 
the  founders  of  modern  geometry. 

Descartes,  Rene,  du  Perron.  Born  at  La  Haye,  Touraine,  1596 ; 
died  at  Stockholm,  1650.  Discoverer  of  analytic  geometry. 
Contributed  extensively  to  algebra. 

Dinostratus.  Lived  about  — 335.  Greek  geometer.  Brother  of 
Menaechmus.  His  name  is  connected  with  the  quadratrix. 

Diodes.  Lived  about  — 180.  Greek  geometer.  Discovered  the 
cissoid  which  he  used  in  solving  the  Delian  problem. 

Diophantus  of  Alexandria.  Lived  about  275.  Most  prominent  of 
Greek  algebraists,  contributing  especially  to  indeterminate 
equations. 


306  HISTORY  OF  MATHEMATICS. 

Dirichlet,  Peter  Gustav  Lejeune.  Born  at  Diiren,  1805  ;  died  at 
Gottingen,  1859.  Succeeded  Gauss  as  professor  at  Gottingen 
Prominent  contributor  to  the  theory  of  numbers. 

Dodson,  James.  Died  Nov.  23,  1757.  Great  grandfather  of  De 
Morgan.  Known  chiefly  for  his  extensive  table  of  anti-log- 
arithms (1742). 

Donatella,  1386-1468.     Italian  sculptor. 

Du  Bois-Reymond,  Paul  David  Gustav.  Born  at  Berlin,  Dec.  2, 
1831  ;  died  at  Freiburg,  April  7,  1889.  Professor  of  mathe- 
matics in  Heidelberg,  Freiburg,  and  Tubingen. 

Duhamel,  Jean  Marie  Constant.  Born  at  Saint-Malo,  1797  ;  died 
at  Paris,  1872.  One  of  the  first  to  write  upon  method  in  math- 
ematics. 

Dupin,  Frangois  Pierre  Charles.  Born  at  Varzy,  1784  ;  died  at 
Paris,  1873. 

Dttrer,  Albrecht.  Born  at  Nuremberg,  1471  ;  died  there,  1528. 
Famous  artist.  One  of  the  founders  of  the  modern  theory  of 
curves. 

Eisenstein,  Ferdinand  Gotthold  Max.  Born  at  Berlin,  1823  ;  died 
there,  1852.  One  of  the  earliest  workers  in  the  field  of  invari 
ants  and  covariants. 

Enneper,  Alfred.     1830-1885.     Professor  at  Gottingen. 

Epaphroditus.  Lived  about  200.  Roman  surveyor.  Wrote  on 
surveying,  theory  of  numbers,  and  mensuration. 

Eratosthenes.  Born  at  Cyrene,  Africa,  — 276  ;  died  at  Alexan- 
dria,— 194.  Prominent  geographer.  Known  for  his  "sieve  " 
for  finding  primes. 

Euclid.  Lived  about  — 300.  Taught  at  Alexandria  in  the  reign 
of  Ptolemy  Soter.  The  author  or  compiler  of  the  most  famous 
text-book  of  Geometry  ever  written,  the  Elements,  in  thirteen 
books. 

Eudoxus  of  Cnidus.  — 408,  — 355.  Pupil  of  Archytas  and  Plato. 
Prominent  geometer,  contributing  especially  to  the  theories  of 
proportion,  similarity,  and  "  the  golden  section." 

Euler,  Leonhard.  Born  at  Basel,  1707  ;  died  at  St.  Petersburg, 
1783.  One  of  the  greatest  physicists,  astronomers  and  math- 
ematicians of  the  i8th  century.  "In  his  voluminous  .  . 


BIOGRAPHICAL  NOTES.  307 

writings  will  be  found  a  perfect  storehouse  of  investigations 
on  every  branch  of  algebraical  and  mechanical  science." — 
Kelland. 

Eutocius.  Born  at  Ascalon,  480.  Geometer.  Wrote  commen- 
taries on  the  works  of  Archimedes,  Apollonius,  and  Ptolemy. 

Fagnano,  Giulio  Carlo,  Count  de.  Born  at  Sinigaglia,  1682  ;  died 
in  1766.  Contributed  to  the  study  of  curves.  Euler  credits 
him  with  the  first  work  in  elliptic  functions. 

Faulhaber,  Johann.     1580-1635.     Contributed   to  the  theory  of 

series. 
Fermat,  Pierre  de.     Born  at  Beaumont-de-Lomagne,  near  Mon- 

tauban,  1601  ;  died  at  Castres,  Jan.  12,  1665.  One  of  the  most 

versatile  mathematicians  of  his  time  ;  his  work  on  the  theory 

of  numbers  has  never  been  equalled. 

Ferrari,  Ludovico.  Born  at  Bologna,  1522  ;  died  in  1562.  Solved 
the  biquadratic. 

Ferro,  Scipione  del.  Born  at  Bologna,  c.  1465  ;  died  between 
Oct.  29  and  Nov.  16,  1526.  Professor  of  mathematics  at  Bo- 
logna. Investigated  the  geometry  based  on  a  single  setting  of 
the  compasses,  and  was  the  first  to  solve  the  special  cubic 
x*+jX  =  q. 

Feuerbach,  Karl  Wilhelm.  Born  at  Jena,  1800 ;  died  in  1834. 
Contributed  to  modern  elementary  geometry. 

Fibonacci.     See  Leonardo  of  Pisa. 

Fourier,  Jean  Baptiste  Joseph,  Baron.  Born  at  Auxerre,  1768 ; 
died  at  Paris,  1830.  Physicist  and  mathematician.  Contrib- 
uted to  the  theories  of  equations  and  of  series. 

Frenicle.     Bernard   Frenide  de   Bessy.     1605-1675.     Friend   of 

Fermat. 
Frezier,  Amede'e  Fra^ois.     Born  at  Chambe'ry,   1682  ;  died  at 

Brest,  1773.     One  of  the  founders  of  descriptive  geometry. 

Friedlein,  Johann  Gottfried.     Born  at  Regensburg,  1828  ;  died  in 

i875. 
Frontinus,  Sextus  Julius.    40-103.    Roman  surveyor  and  engineer. 

Galois,  Evariste.  Born  at  Paris,  1811  ;  died  there,  1832.  Founder 
of  the  theory  of  groups. 


308  HISTORY  OF  MATHEMATICS. 

Gauss,  Karl  Friedrich.  Born  at  Brunswick,  1777;  died  at  Got- 
tingen,  1855.  The  greatest  mathematician  of  modern  times. 
Prominent  as  a  physicist  and  astronomer.  The  theories  of 
numbers,  of  functions,  of  equations,  of  determinants,  of  com- 
plex numbers,  of  hyperbolic  geometry,  are  all  largely  indebted 
to  his  great  genius. 

Geber.  Jabir  ben  Aflah.  Lived  about  1085.  Astronomer  at  Se- 
ville ;  wrote  on  spherical  trigonometry. 

Gettibrand,  Henry.  1597-1637.  Prof essor  of  astronomy  at  Gresham 
College. 

Geminus.  Born  at  Rhodes,  — 100 ;  died  at  Rome,  — 40.  Wrote 
on  astronomy  and  (probably)  on  the  history  of  pre-Euclidean 
mathematics. 

Gerbert,  Pope  Sylvester  II.  Born  at  Auvergne,  940 ;  died  at 
Rome,  May  13,  1003.  Celebrated  teacher  ;  elected  pope  in 
999.  Wrote  upon  arithmetic. 

Gerhard  of  Cremona.  From  Cremona  (or,  according  to  others, 
Carmona  in  Andalusia).  Born  in  1114  ;  died  at  Toledo  in 
1187.  Physician,  mathematician,  and  astrologer.  Translated 
several  works  of  the  Greek  and  Arab  mathematicians  from 
Arabic  into  Latin. 

Germain,  Sophie.     1776-1831.     Wrote  on  elastic  surfaces. 

Girard,  Albert,  c.  1590-1633.  Contributed  to  the  theory  of  equa- 
tions, general  polygons,  and  symbolism. 

Gopel,  Gustav  Adolf.     1812-1847.     Known  for  his  researches  on 

hyperelliptic  functions. 
Grammateus,   Henricus.     (German  name,    Heinrich  Schreiber.) 

Born  at  Erfurt,  c.  1476.     Arithmetician. 

Grassmann,  Hermann  Gunther.  Born  at  Stettin,  April  15,  1809  : 
died  there  Sept.  26,  1877.  Chiefly  known  for  his  Aiisdehmui.ifx- 
lehre  (1844).  Also  wrote  on  arithmetic,  trigonometry,  and 
physics. 

Grebe,  Ernst  Wilhelm.  Born  near  Marbach,  Oberhesse,  Aug.  30. 
1804  ;  died  at  Cassel,  Jan.  14,  1874.  Contributed  to  modern 
elementary  geometry. 

Gregory,  James.  Born  at  Drumoak,  Aberdeenshire,  Nov.  1638  ; 
died  at  Edinburgh,  1675.  Professor  of  mathematics  at  St.  An- 


BIOGRAPHICAL  NOTES.  309 

drews  and  Edinburgh.     Proved  the  incommensurability  of  rr ; 
contributed  to  the  theory  of  series. 

Grunert,  Johann  August.  Born  at  Halle  a.  S.,  1797;  died  in  1872 
Professor  at  Greifswalde,  and  editor  of  Grunert's  Archiv. 

Gua.  Jean  Paul  de  Gua  de  Malves.  Born  at  Carcassonne,  1713  ; 
died  at  Paris,  June  2,  1785.  Gave  the  first  rigid  proof  of  Des- 
cartes's  rule  of  signs. 

Gudermann,  Christoph.  Born  at  Winneburg,  March  28,  1798  ; 
died  at  Miinster,  Sept.  25,  1852.  To  him  is  largely  due  the 
introduction  of  hyperbolic  functions  into  modern  analysis. 

Guldin,  Habakkuk  (Paul).  Born  at  St.  Gall,  1577;  died  at  Gratz, 
1643.  Known  chiefly  for  his  theorem  on  a  solid  of  revolution, 
pilfered  from  Pappus. 

Hachette,  Jean  Nicolas  Pierre.  Born  at  Me'zieres,  1769 ;  died  at 
Paris,  1834.  Algebraist  and  geometer. 

Hattey,  Edmund.  Born  at  Haggerston,  near  London,  Nov.  8, 
1656  ;  died  at  Greenwich,  Jan.  14,  1742.  Chiefly  known  for 
his  valuable  contributions  to  physics  and  astronomy. 

Halfhen,  George  Henri.  Born  at  Rouen,  Oct.  30,  1844  ;  died  at 
Versailles  in  1889.  Professor  in  the  Ecole  Polytechnique  at 
Paris.  Contributed  to  the  theories  of  differential  equations 
and  of  elliptic  functions. 

Hamilton,  Sir  William  Rowan.  Born  at  Dublin,  Aug.  3-4,  1805  ; 
died  there,  Sept.  2,  1865.  Professor  of  astronomy  at  Dublin. 
Contributed  extensively  to  the  theory  of  light  and  to  dynamics, 
but  known  generally  for  his  discovery  of  quaternions. 

Hankel,  Hermann.  Born  at  Halle,  Feb.  14,  1839 ;  died  at  Schram- 
berg,  Aug.  29,  1873.  Contributed  chiefly  to  the  theory  of  com- 
plex numbers  and  to  the  history  of  mathematics. 

Ifarnack,  Karl  Gustav  Axel.  Born  at  Dorpat,  1851;  died  at  Dres- 
den in  1888.  Professor  in  the  polytechnic  school  at  Dresden. 

Harriot,  Thomas.  Born  at  Oxford,  1560 ;  died  at  Sion  House, 
near  Isleworth,  July  2,  1621.  The  most  celebrated  English 
algebraist  of  his  time. 

Heron  of  Alexandria.  Lived  about  — no.  Celebrated  surveyor 
and  mechanician.  Contributed  to  mensuration. 


310  HISTORY  OF  MATHEMATICS. 

Hesse,  Ludwig  Otto.  Born  at  KOnigsberg,  April  22,  1811 ;  died 
at  Munich,  Aug.  4,  1874.  Contributed  to  the  theories  of  curves 
and  of  determinants. 

Hipparchus.  Born  at  Nicaea,  Bithynia,  — 180 ;  died  at  Rhodes, 
— 125.  Celebrated  astronomer.  One  of  the  earliest  writers 
on  spherical  trigonometry. 

Hippias  of  Elis.  Born  c.  — 460.  Mathematician,  astronomer, 
natural  scientist.  Discovered  the  quadratrix. 

Hippocrates  of  Chios.  Lived  about  — 440.  Wrote  the  first  Greek 
elementary  text-book  on  mathematics. 

Homer,  William  George.  Born  in  1786  ;  died  at  Bath,  Sept.  22, 
1837.  Chiefly  known  for  his  method  of  approximating  the  real 
roots  of  a  numerical  equation  (1819). 

Hrabanus  Maurus.     788-856.     Teacher  of  mathematics.     Arch 

bishop  of  Mainz. 
Hudde,   Johann.     Born  at  Amsterdam,    1633;  died  there,   1704 

Contributed  to  the  theories  of  equations  and  of  series. 

Honein  ibn  Ishak.  Died  in  873.  Arab  physician.  Translated 
several  Greek  scientific  works. 

Huygens,  Christiaan,  van  Zuylichem.  Born  at  the  Hague,  1629  ; 
died  there,  1695.  Famous  physicist  and  astronomer.  In  math- 
ematics he  contributed  to  the  study  of  curves. 

Hyginus.     Lived  about  100.     Roman  surveyor. 

Hypatia,  daughter  of  Theon  of  Alexandria.  375-415.  Composed 
several  mathematical  works.  See  Charles  Kingsley's  Hypatia. 

Hypsicles  of  Alexandria.  Lived  about  —190.  Wrote  on  solid 
geometry  and  theory  of  numbers,  and  solved  certain  indeter- 
minate equations. 

lamblichus.  Lived  about  325.  From  Chalcis.  Wrote  on  various 
branches  of  mathematics. 

Ibn  al  Banna.  Abul  Abbas  Ahmed  ibn  Mohammed  ibn  Otman  al 
Azdi  al  Marrakushi  ibn  al  Banna  Algarnati.  Born  1252  or 
1257  in  Morocco.  West  Arab  algebraist;  prolific  writer. 

7bn  Yunus,  Abul  Hasan  Ali  ibn  Abi  Said  Abderrahman.  960 
1008.  Arab  astronomer  ;  prepared  the  Hakimitic  Tables. 


BIOGRAPHICAL  NOTES.  31! 

Isidorus  Hispalensis.  Born  at  Carthagena,  570 ;  died  at  Seville, 
636.  Bishop  of  Seville.  His  Origines  contained  dissertations 
on  mathematics. 

Ivory,  James.  Born  at  Dundee,  1765 ;  died  at  London,  Sept.  21, 
1842.  Chiefly  known  as  a  physicist. 

Jacobi,  Karl  Gustav  Jacob.  Born  at  Potsdam,  Dec.  10,  1804; 
died  at  Berlin,  Feb.  18,  1851.  Important  contributor  to  the 
theory  of  elliptic  and  theta  functions  and  to  that  of  functional 
determinants. 

Jamin,  Jules  Ce"lestm.  Born  in  1818 ;  died  at  Paris,  1886.  Pro- 
fessor of  physics. 

Joannes  de  Praga  (Johannes  Schindel).  Born  at  KSniggratz,  1370 
or  1375  ;  died  at  Prag  c.  1450.  Astronomer  and  mathema- 
tician. 

Johannes  of  Seville  (Johannes  von  Luna,  Johannes  Hispalensis). 
Lived  about  1140.  A  Spanish  Jew;  wrote  on  arithmetic  and 
algebra. 

Johann  von  Gmttnden.  Born  at  Gmtinden  am  Traunsee,  between 
1375  and  1385  ;  died  at  Vienna,  Feb.  23,  1442.  Professor  of 
mathematics  and  astronomy  at  Vienna ;  the  first  full  professor 
of  mathematics  in  a  Teutonic  university. 

Kdstner,  Abraham  Gotthelf.  Born  at  Leipzig,  1719;  died  at  G6t- 
tingen,  1800.  Wrote  on  the  history  of  mathematics. 

Kepler,  Johann.  Born  in  Wtirtemberg,  near  Stuttgart,  1571 ;  died 
at  Regensburg,  1630.  Astronomer  (assistant  of  Tycho  Brahe, 
as  a  young  man);  "may  be  said  to  have  constructed  the  edi- 
fice of  the  universe," — Proctor.  Prominent  in  introducing  the 
use  of  logarithms.  Laid  down  the  "principle  of  continuity" 
(1604);  helped  to  lay  the  foundation  of  the  infinitesimal  cal- 
culus. 

Khayyam,  Omar.  Died  at  Nishapur,  1123.  Astronomer,  geometer, 
algebraist.  Popularly  known  for  his  famous  collection  of 
quatrains,  the  Rubaiyat. 

KSbel,  Jacob.  Born  at  Heidelberg,  1470 ;  died  at  Oppenheim,  in 
1533.  Prominent  writer  on  arithmetic  (1514,  1520). 

Lacroix,  Sylvestre  Franjois.  Born  at  Paris,  1765 ;  died  there, 
May  25,  1843.  Author  of  an  elaborate  course  of  mathematics. 


312  HISTORY  OF  MATHEMATICS. 

Laguerre,  Edmond  Nicolas.  Born  at  Bar-le-Duc,  April  9,  1834 ; 
died  there  Aug.  14,  1886.  Contributed  to  higher  analysis. 

Lagrange,  Joseph  Louis,  Comte.  Born  at  Turin,  Jan.  25,  1736; 
died  at  Paris,  April  10,  1813.  One  of  the  foremost  mathe- 
maticians of  his  time.  Contributed  extensively  to  the  calculus 
of  variations,  theory  of  numbers,  determinants,  differential 
equations,  calculus  of  finite  differences,  theory  of  equations, 
and  elliptic  functions.  Author  of  the  Mgcanique  analytique. 
Also  celebrated  as  an  astronomer. 

Lahire,  Philippe  de.  Born  at  Paris,  March  18,  1640;  died  there 
April  21,  1718.  Contributed  to  the  study  of  curves  and  magic 
squares. 

Laloubere,  Antoine  de.  Born  in  Languedoc,  1600;  died  at  Tou- 
louse, 1664.  Contributed  to  the  study  of  curves. 

Lambert,  Johann  Heinrich.  Born  at  Mulhausen,  Upper  Alsace, 
1728 ;  died  at  Berlin,  1777.  Founder  of  the  hyperbolic  trigo- 
nometry. 

Lame,  Gabriel.  Born  at  Tours,  1795  ;  died  at  Paris,  1870.  Writer 
on  elasticity ,  and  orthogonal  surfaces. 

Landen,  John.  Born  at  Peakirk,  near  Peterborough,  1719 ;  died 
at  Milton,  1790.  A  theorem  of  his  (1755)  suggested  to  Euler 
and  Lagrange  their  study  of  elliptic  integrals. 

Laplace,  Pierre  Simon,  Marquis  de.  Born  at  Beaumont-en-Auge, 
Normandy,  March  23,  1749;  died  at  Paris,  March  5,  1827. 
Celebrated  astronomer,  physicist,  and  mathematician.  Added 
to  the  theories  of  least  squares,  determinants,  equations,  se- 
ries, probabilities,  and  differential  equations. 

Legendre,  Adrien  Marie.  Born  at  Toulouse,  Sept.  18,  1752 ;  died 
at  Paris,  Jan.  10,  1833.  Celebrated  mathematician,  contribut- 
ing especially  to  the  theory  of  elliptic  functions,  theory  of 
numbers,  least  squares,  and  geometry.  Discovered  the  "law 
of  quadratic  reciprocity," — "the  gem  of  arithmetic"  (Gauss). 

Leibnitz,  Gottfried  Wilhelm.  Born  at  Leipzig,  1646;  died  at 
Hanover  in  1716.  One  of  the  broadest  scholars  of  modern 
times;  equally  eminent  as  a  philosopher  and  mathematician. 
One  of  the  discoverers  of  the  infinitesimal  calculus,  and  the 
inventor  of  its  accepted  symbolism. 


BIOGRAPHICAL  NOTES. 


313 


Leonardo  of  Pisa,  Fibonacci  (filius  Bonacii,  son  of  Bonacius). 
Born  at  Pisa,  1180;  died  in  1250.  Travelled  extensively  and 
brought  back  to  Italy  a  knowledge  of  the  Hindu  numerals  and 
the  general  learning  of  the  Arabs,  which  he  set  forth  in  his 
Liber  Abaci,  Practica  geometriae,  and  Flos. 

[.'Hospital,  Guillaume  Fran9ois  Antoine  de,  Marquis  de  St. 
Mesme.  Born  at  Paris,  1661 ;  died  there  1704.  One  of  the 
first  to  recognise  the  value  of  the  infinitesimal  calculus. 

Lhuilier,  Simon  Antoine  Jean.     Born  at  Geneva,   1750;  died  in 

1840.     Geometer. 
Libri,  Carucci  dalla  Sommaja,  Guglielmo  Brutus  Icilius  Timoleon. 

Born  at  Florence,  Jan.  2,  1803 ;  died  at  Villa  Fiesole,  Sept. 

28,  1869.     Wrote  on  the  history  of  mathematics  in  Italy. 

Lie,  Marius  Sophus.  Born  Dec.  12,  1842;  died  £eb.  18,  1899. 
Professor  of  mathematics  in  Christiania  and  Leipzig.  Spe- 
cially celebrated  for  his  theory  of  continuous  groups  of  trans- 
formations as  applied  to  differential  equations. 

Liouville,  Joseph.  Born  at  St.  Omer,  1809  ;  died  in  1882.  Founder 
of  the  journal  that  bears  his  name. 

Lobachevsky,  Nicolai  Ivanovich.  Born  at  Makarief,  1793;  died 
at  Kasan,  Feb.  12-24,  1856.  One  of  the  founders  of  the  so- 
called  non-Euclidean  geometry. 

Ludolph  van  Ceulen.     See  Van  Ceulen. 

MacCullagh,  James.  Born  near  Strabane,  1809;  died  at  Dublin, 
1846.  Professor  of  mathematics  and  physics  in  Trinity  Col- 
lege, Dublin. 

Maclaurin,  Colin.  Born  at  Kilmodan,  Argyllshire,  1698 ;  died  at 
York,  June  14,  1746.  Professor  of  mathematics  at  Edinburgh. 
Contributed  to  the  study  of  conies  and  series 

Malfatti,  Giovanni  Francesco  Giuseppe.  Born  at  Ala,  Sept.  26, 
1731 ;  died  at  Ferrara,  Oct.  9,  1807.  Known  for  the  geomet- 
ric problem  which  bears  his  name, 

Malus,  Etienne  Louis.  Born  at  Paris,  June  23,  1775 ;  died  there, 
Feb.  24,  1812.  Physicist. 

\h\sfheroni,  Lorenzo.  Born  at  Castagneta,  1750;  died  at  Paris, 
r8oo.  First  to  elaborate  the  geometry  of  the  compasses  only 
(1795). 


314  HISTORY  OF  MATHEMATICS. 

Maurolico,  Francesco.  Born  at  Messina,  Sept.  16,  1494;  died 
July  21,  1575.  The  leading  geometer  of  bis  time.  Wrote  also 
on  trigonometry. 

Maximus  Planudes.  Lived  about  1330.  From  Nicomedia.  Greek 
mathematician  at  Constantinople.  \  rote  a  commentary  on 
Diophantus ;  also  on  arithmetic. 

Menaechmus.  Lived  about  — 350.  Pupil  of  Plato.  Discoverer 
of  the  conic  sections. 

Menelaus  of  Alexandria.  Lived  about  100.  Greek  mathematician 
and  astronomer.  Wrote  on  geometry  and  trigonometry. 

Mercator,  Gerhard.  Born  at  Rupelmonde,  Flanders,  1512 :  died 
at  Duisburg,  1594.  Geographer. 

Mercator,  Nicholas.  (German  name  Kaufmann.)  Born  near 
Cismar,  Holstein,  c.  1620;  died  at  Paris,  1687.  Discovered 
the  series  for  log  (1  +*). 

Metius,  Adriaan.  Born  at  Alkmaar,  1571  ;  died  at  Franeker,  1635 
Suggested  an  approximation  for  TT,  really  due  to  his  father. 

Meusnier  de  la  Place,  Jean  Baptiste  Marie  Charles.  Born  at 
Paris,  1754  ;  died  at  Cassel,  1793.  Contributed  a  theorem  on 
the  curvature  of  surfaces. 

Mgziriac,  Claude  Gaspard  Bachet  de.  Born  at  Bourg-en-Bresse, 
1581 ;  died  in  1638.  Known  for  his  Problemes  plaisants,  etc. 
(1624)  and  his  translation  of  Diophantus. 

Mobius,  August  Ferdinand.  Born  at  Schulpforta,  Nov.  17,  1790  ; 
died  at  Leipzig,  Sept.  26,  1868.  One  of  the  leaders  in  modern 
geometry.  Author  of  Der  Barycentrische  Calciil  (1827) . 

Mohammed  ibn  Musa.     See  Al  Khowarazmi. 
Moivre.     See  DeMoivre. 

Moll-weide,  Karl  Brandan.  Born  at  Wolfenbtittel,  Feb.  3,  1774  ; 
died  at  Leipzig,  March  10,  1825.  Wrote  on  astronomy  and 
mathematics. 

Monge,  Gaspard,  Comte  de  Peluse.  Born  at  Beaune,  1746  ;  died 
at  Paris,  1818.  Discoverer  of  descriptive  geometry;  contrib- 
uted to  the  study  of  curves  and  surfaces,  and  to  differential 
equations. 


BIOGRAPHICAL  NOTES.  315 

Montmort,  Pierre  Esmond  de.  Born  at  Paris,  1678  ;  died  there, 
1719.  Contributed  to  the  theory  of  probabilities  and  to  the 
summation  of  series. 

Moschopulus,  Manuel.  Lived  about  1300.  Byzantine  mathemati- 
cian. Known  for  his  work  on  magic  squares. 

Mydorge,  Claude.  Born  at  Paris,  1585  ;  died  there  in  1647.  Author 
of  the  first  French  treatise  on  conies. 

Napier,  John.  Born  at  Merchiston,  then  a  suburb  of  Edinburgh, 
1550  ;  died  there  in.  1617.  Inventor  of  logarithms.  Contrib- 
uted to  trigonometry. 

Newton,  Sir  Isaac.  Born  at  Woolsthorpe,  Lincolnshire,  Dec.  25, 
1642,  O.  S. ;  died  at  Kensington,  March  20,  1727.  Succeeded 
Barrow  as  Lucasian  professor  of  mathematics  at  Cambridge 
(1669).  The  world's  greatest  mathematical  physicist.  Invented 
fluxional  calculus  (c.  1666).  Contributed  extensively  to  the 
theories  of  series,  equations,  curves,  and,  in  general,  to  all 
branches  of  mathematics  then  known. 

Nicole,  Francois.  Born  at  Paris,  1683 ;  died  there,  1758.  First 
treatise  on  finite  differences. 

Nicomachus  of  Gerasa,  Arabia.  Lived  100.  Wrote  upon  arith- 
metic. 

Nicomedes  of  Gerasa.  Lived  — 180.  Discovered  the  conchoid 
which  bears  his  name. 

Nicolaus  von  Cusa.  Born  at  Cuss  on  the  Mosel,  1401  ;  died  at 
Todi,  Aug.  ii,  1464.  Theologian,  physicist,  astronomer,  ge- 
ometer. 

Odo  of  Cluny.  Born  at  Tours.  879  ;  died  at  Cluny,  942  or  943. 
Wrote  on  arithmetic. 

Oenopides  of  Chios.     Lived  — 465.    Studied  in  Egypt.    Geometer. 

Olivier,  Theodore.  Born  at  Lyons,  Jan.  21,  1793  ;  died  in  same 
place  Aug.  5,  1853.  Writer  on  descriptive  geometry. 

Oresme,  Nicole.  Born  in  Normandy,  c.  1320;  died  at  Lisieux, 
1382.  Wrote  on  arithmetic  and  geometry. 

Oughtred,  William.  Born  at  Eton,  1574  ;  died  at  Albury,  1660. 
Writer  on  arithmetic  and  trigonometry. 

Pacioli,  Luca.  Fra  Luca  di  Borgo  di  Santi  Sepulchri.  Born  at 
Borgo  San  Sepolcro,  Tuscany,  c.  1445  ;  died  at  Florence, 


316  HISTORY  OF  MATHEMATICS. 

c.  1509.     Taught  in  several  Italian  cities.     His  Summa  de 

Arithmetica,  Geometria,  etc.,  was  the  first  great  mathemat 

ical  work  published  (1494). 
Pappus  of  Alexandria.     Lived  about  300.     Compiled  a  work  con 

taining  the  mathematical  knowledge  of  his  time. 
Parent,  Antoine.     Born  at  Paris,  1666;  died  there  in  1716.    Fiist 

to  refer  a  surface  to  three  co-ordinate  planes  (1700). 
Pascal,  Blaise.     Born  at  Clermont,    1623;  died  at  Paris,   1662 

Physicist,   philosopher,    mathematician.     Contributed   to  the 

theory  of  numbers,  probabilities,  and  geometry. 
Peirce,  Charles  S.     Born  at  Cambridge,   Mass.,  Sept.   10,   1839 

Writer  on  logic. 
Pell,  John.    Born  in  Sussex,  March  i,  1610 ;  died  at  London,  Dec 

10,  1685.     Translated  Rahn's  algebra. 

Perseus.     Lived  — 150.     Greek  geometer  ;  studied  spiric  lines. 
Peuerbach,  Georg  von.     Born  at  Peuerbach,  Upper  Austria,  May 

30,  1423;  died  at  Vienna,  April  8,  1461.     Prominent  teacher 

and  writer  on  arithmetic,  trigonometry,  and  astronomy. 
Pfaff,  Johann  Friedrich.     Born  at  Stuttgart,  1765  ;  died  at  Halle 

in  1825.     Astronomer  and  mathematician. 

Pitiscus,  Bartholomaeus.     Born  Aug.   24,   1561  ;  died  at  Heidel- 
berg, July  2,  1613.    Wrote  on  trigonometry,  and  first  used  the 

present  decimal  point  (1612). 
Plana,  Giovanni   Antonio   Amedeo.     Born  at  Voghera,  Nov.  8, 

1781;  died  at  Turin,  Jan.  2,  1864.     Mathematical  astronomer 

and  physicist. 

Planudes.     See  Maximus  Planndes. 
Plateau,  Joseph  Antoine  Ferdinand.     Born  at  Brussels,  Oct.  14, 

1801  ;  died  at  Ghent,  Sept.  15,  1883.     Professor  of  physics  at 

Ghent. 
Plato.     Born  at  Athens,  — 429;  died  in  — 348.     Founder  of  the 

Academy.     Contributed  to  the  philosophy  of  mathematics. 
Plato  of  Tivoli.     Lived  1120.     Translated  Al  Battani's  trigonom- 
etry and  other  works. 
Plilcker,  Johann.    Born  at  Elberfeld,  July  16,  1801  ;  died  at  Bonn, 

May  22,  1868.     Professor  of  mathematics  at  Bonn  and  Halle. 

One  of  the  foremost  geometers  of  the  century. 


BIOGRAPHICAL  NOTES.  317 

Poisson,  Simeon  Denis.  Born  at  Pithiviers,  Loiret,  1781  ;  died 
at  Paris,  1840.  Chiefly  known  as  a  physicist.  Contributed 
to  the  study  of  definite  integrals  and  of  series. 

Poncelet,  Jean  Victor.  Born  at  Metz,  1788  ;  died  at  Paris,  1867. 
One  of  the  founders  of  projective  geometry. 

Pothenot,  Laurent.  Died  at  Paris  in  1732.  Professor  of  mathe- 
matics in  the  College  Royale  de  France. 

Proclus.  Born  at  Byzantium,  412;  died  in  485.  Wrote  a  com- 
mentary on  Euclid.  Studied  higher  plane  curves. 

Ptolemy  (Ptolemaeus  Claudius).  Born  at  Ptolemais,  87;  died  at 
Alexandria,  165.  One  of  the  greatest  Greek  astronomers. 

Pythagoras.  Born  at  Samos,  — 580  ;  died  at  Megapontum,  — 501. 
Studied  in  Egypt  and  the  East.  Founded  the  Pythagorean 
school  at  Croton,  Southern  Italy.  Beginning  of  the  theory  of 
numbers.  Celebrated  geometrician. 

Quetelet,  Lambert  Adolph  Jacques.  Born  at  Ghent,  Feb.  22, 
1796 ;  died  at  Brussels,  Feb.  7,  1874.  Director  of  the  royal 
observatory  of  Belgium.  Contributed  to  geometry,  astronomy, 
and  statistics. 

Ramus,  Peter  (Pierre  de  la  Ramee).  Born  at  Cuth,  Picardy,  1515  ; 
murdered  at  the  massacre  of  St.  Bartholomew,  Paris,  August 
24-25,  1572.  Philosopher,  but  also  a  prominent  writer  on 
mathematics. 

Recorde,  Robert.  Born  at  Tenby,  Wales,  1510;  died  in  prison,, 
at  London,  1558.  Professor  of  mathematics  and  rhetoric  at 
Oxford.  Introduced  the  sign  =  for  equality. 

Rcgiomontanus.  Johannes  Muller.  Born  near  KSnigsberg,  June 
6,  1436  ;  died  at  Rome,  July  6,  1476.  Mathematician,  astron- 
omer, geographer.  Translator  of  Greek  mathematics.  Author 
of  first  text-book  of  trigonometry. 

Rcmigius  of  Auxerre.  Died  about  908.  Pupil  of  Alcuin's.  Wrote 
on  arithmetic. 

Rhaeticus,  Georg  Joachim.  Born  at  Feldkirch,  1514;  died  at 
Kaschau,  1576.  Professor  of  mathematics  at  Wittenberg  ;  pu- 
pil of  Copernicus  and  editor  of  his  works.  Contributed  to 
trigonometry. 


318  HISTORY  OF  MATHEMATICS. 

Riccati,  Count  Jacopo  Francesco.  Born  at  Venice,  1676 ;  died  at 
Treves,  1754.  Contributed  to  physics  and  differential  equa- 
tions. 

Richelot,  Friedrich  Julius.  Born  at  Konigsberg,  Nov.  6,  1808  ; 
died  March  31,  1875  in  same  place.  Wrote  on  elliptic  and 
Abelian  functions. 

Riemann,  George  Friedrich  Bernhard.  Born  at  Breselenz,  Sept. 
17,  1826 ;  died  at  Selasca,  July  20,  1866.  Contributed  to  the 
theory  of  functions  and  to  the  study  of  surfaces. 

Riese,  Adam.  Born  at  Staffelstein,  near  Lichtenfels,  1492  ;  died 
at  Annaberg,  1559.  Most  influential  teacher  of  and  writer  on 
arithmetic  in  the  i6th  century. 

Roberval,  Giles  Persone  de.  Born  at  Roberval,  1602  ;  died  at 
Paris,  1675.  Professor  of  mathematics  at  Paris.  Geometry 
of  tangents  and  the  cycloid. 

Rotte,  Michel.  Born  at  Ambert,  April  22,  1652  ;  died  at  Paris, 
Nov.  8,  1719.  Discovered  the  theorem  which  bears  his  name, 
in  the  theory  of  equations. 

Rudolff,  Christoff.  Lived  in  first  part  of  the  sixteenth  century. 
German  algebraist. 

Sacro-Bosco,  Johannes  de.  Born  at  Holywood  (Halifax),  York- 
shire, i2Oo(?);  died  at  Paris,  1256.  Professor  of  mathematics 
and  astronomy  at  Paris.  Wrote  on  arithmetic  and  trigonom- 
etry. 

Saint-Venant,  Adhemar  Jean  Claude  Barr£  de.  Born  in  1797  ; 
died  in  Vendome,  1886.  Writer  on  elasticity  and  torsion. 

Saint-Vincent,  Gregoire  de.  Born  at  Bruges,  1584  ;  died  at  Ghent, 
1667.  Known  for  his  vain  attempts  at  circle  squaring. 

Saurin,  Joseph.     Born  at  Courtaison,  1659;  died  at  Paris,  1737. 

Geometry  of  tangents. 
Scheeffer,  Ludwig.     Born  at  Konigsberg,  1859 ;  died  at  Munich, 

1885.     Writer  on  theory  of  functions. 
Schindel,  Johannes.     See  Joannes  de  Praga. 

Schzuenter,  Daniel.  Born  at  Nuremberg,  1585  ;  died  in  1636. 
Professor  of  oriental  languages  and  of  mathematics  at  Altdorf. 

Serenus  of  Antissa.     Lived  about  350.     Geometer. 


BIOGRAPHICAL  NOTES.  319 

Serret,  Joseph  Alfred.  Born  at  Paris,  Aug.  30,  1819  ;  died  at 
Versailles,  March  2,  1885.  Author  of  well-known  text-books 
on  algebra  and  the  differential  and  integral  calculus. 

Sextus  Julius  Africanus.  Lived  about  220.  Wrote  on  the  his- 
tory of  mathematics. 

Simpson,  Thomas.  Born  at  Bosworth,  Aug.  20,  1710 ;  died  at 
Woolwich,  May  14,  1761.  Author  of  text-books  on  algebra, 
geometry,  trigonometry,  and  fluxions. 

Sluze,  Rene  Fran§ois  Walter  de.  Born  at  Vis£  on  the  Maas,  1622  ; 
died  at  Liege  in  1685.  Contributed  to  the  notation  of  the  cal- 
culus, and  to  geometry. 

Smith,  Henry  John  Stephen.  Born  at  Dublin,  1826 ;  died  at  Ox- 
ford, Feb.  9,  1883.  Leading  English  writer  on  theory  of  num- 
bers. 

Snell,  Willebrord,  van  Roijen.  Born  at  Leyden,  1591  ;  died  there, 
1626.  Physicist,  astronomer,  and  contributor  to  trigonometry. 

Spottisuuoode,  William.  Born  in  London,  Jan.  n,  1825  ;  died 
there,  June  27,  1883.  President  of  the  Royal  Society.  Writer 
on  algebra  and  geometry. 

Staudt,  Karl  Georg  Christian  von.  Born  at  Rothenburg  a.  d. 
Tauber,  Jan.  24,  1798  ;  died  at  Erlangen,  June  i,  1867.  Prom- 
inent contributor  to  modern  geometry,  Geometrie  der  Lage. 

Steiner,   Jacob.     Born  at  Utzendorf,   March   18,    1796 ;    died  at 

Bern,  April  i,  1863.     Famous  geometrician. 
Stevin,  Simon.     Born  at  Bruges,   1548  ;  died  at  Leyden  (or  the 

Hague),  1620.     Physicist  and  arithmetician. 

Stewart,  Matthew.  Born  at  Rothsay,  Isle  of  Bute,  1717;  died  at 
Edinburgh,  1785.  Succeeded  Maclaurin  as  professor  of  math- 
ematics at  Edinburgh.  Contributed  to  modern  elementary 
geometry. 

Stifel,  Michael.  Born  at  Esslingen,  1486  or  1487;  died  at  Jena, 
1567.  Chiefly  known  for  his  Arithmetica  integra  (1544). 

Sturm,  Jacques  Charles  Francois.  Born  in  Geneva,  1803  ;  died 
in  1855.  Professor  in  the  Ecole  Polytechnique  at  Paris. 
"Sturm's  theorem." 

Sylvester,  James  Joseph.  Born  in  London,  Sept.  3,  1814  ;  died 
in  same  place,  March  15,  1897.  Savilian  professor  of  pure 


32O  HISTORY  OF  MATHEMATICS. 

geometry  in  the  University  of  Oxford.     Writer  on  algebra, 
especially  the  theory  of  invariants  and  covariants. 

Tdbit  ibn  Kurra.  Born  at  Harran  in  Mesopotamia,  833  ;  died  at 
Bagdad,  902.  Mathematician  and  astronomer.  Translated 
works  of  the  Greek  mathematicians,  and  wrote  on  the  theory 
of  numbers. 

TartagHa,  Nicolo.  (Nicholas  the  Stammerer.  Real  name,  Ni- 
colo  Fontana.)  Born  at  Brescia,  c.  1500;  died  at  Venice,  c. 
1557.  Physicist  and  arithmetician  ;  best  known  for  his  work 
on  cubic  equations. 

Taylor,  Brook.  Born  at  Edmonton,  1685  ;  died  at  London,  1731. 
Physicist  and  mathematician.  Known  chiefly  for  his  work  in 
series. 

Tholes.  Born  at  Miletus,  — 640 ;  died  at  Athens,  — 548.  One  of 
the  "  seven  wise  men  "  of  Greece  ;  founded  the  Ionian  School. 
Traveled  in  Egypt  and  there  learned  astronomy  and  geom- 
etry. First  scientific  geometry  in  Greece. 

Theaetetus  of  Heraclea.  Lived  in  — 390.  Pupil  of  Socrates. 
Wrote  on  irrational  numbers  and  on  geometry. 

Theodorus  of  Cyrene.  Lived  in  — 410.  Plato's  mathematical 
teacher.  Wrote  on  irrational  numbers. 

Theon  of  Alexandria.  Lived  in  370.  Teacher  at  Alexandria. 
Edited  works  of  Greek  mathematicians. 

Theon  of  Smyrna  Lived  in  130.  Platonic  philosopher.  Wrote 
on  arithmetic,  geometry,  mathematical  history,  and  astronomy. 

Thymaridas  of  Paros.     Lived  in  — 390.     Pythagorean  ;  wrote  on 

arithmetic  and  equations. 
Torricelli,   Evangelista.     Born  at  Faeflza,    1608 ;    died  in   1647. 

Famous  physicist. 

Tortolini,  Barnaba.  Born  at  Rome,  Nov.  19,  1808  ;  died  August 
24,  1874.  Editor  of  the  Annali  which  bear  his  name. 

Trembley,  Jean.     Born  at  Geneva,   1749;  died  in   1811.     Wrote 

on  differential  equations. 
Tschirnhausen,  Ehrenfried  Walter,  Graf  von.     Born  at  Kiess- 

lingswalde,  1651;  died  at  Dresden,  1708.     Founded  the  theory 

of  catacaustics. 


BIOGRAPHICAL  NOTES.  $21 

Ubaldi,  Guido.     See  Del  Monte. 

Unger,  Ephraim  Solomon.     Born  at  Coswig,  1788  ;  died  in  1870. 

Ursinus,  Benjamin.     1587 — 1633.     Wrote  on  trigonometry  and 

computed  tables. 
Van  Ceulen,  Ludolph.    Born  at  HildesheSm,  Jan.  18  (or  28),  1540 ; 

died  in  Holland,  Dec.  31,  1610.     Known  for  his  computations 

Of    7T. 

Vandermonde,  Charles  Auguste.  Born  at  Paris,  in  1735  ;  died 
there,  1796.  Director  of  the  Conservatoire  pour  les  arts  et 
metiers. 

Van  Eyck,  Jan.     1385-1440.     Dutch  painter. 

Van  Schooten,  Franciscus  (the  younger).  Born  in  1615  ;  died  in 
1660.  Editor  of  Descartes  and  Vieta. 

Vitte  (Vieta),  Franjois,  Seigneur  de  la  Bigotiere.  Born  at  Fonte- 
nay-le-Comte,  1540;  died  at  Paris,  1603.  The  foremost  alge- 
braist of  his  time.  Also  wrote  on  trigonometry  and  geometry. 

Vincent.     See  Saint-Vincent. 

Vitruvitis.  Marcus  Vitruvius  Pollio.  Lived  in  — 15.  Roman  archi- 
tect. Wrote  upon  applied  mathematics. 

Viviani,  Vincenzo.     Born  at   Florence,   1622  ;  died  there,   1703. 

Pupil  of  Galileo  and  Torricelli.     Contributed  to  elementary 

geometry. 
Wallace,  William.     Born  in   1768;   died  in  1843.     Professor  of 

mathematics  at  Edinburgh. 
Wallis,  John.     Born  at  Ashford,  1616  ;  died  at  Oxford,  1703.    Sa- 

vilian   professor   of    geometry   at   Oxford.     Published  many 

mathematical  works.     Suggested  (1685)  the  modern  graphic 

interpretation  of  the  imaginary. 
Weierstrass,  Karl  Theodor  Wilhelm.     Born  at  Ostenfelde,  Oct. 

31,  1815  ;  died  at  Berlin,  Feb.  19,  1897.     OQe  °f  *he  ablest 

mathematicians  of  the  century. 
Werner,  Johann.  Born  at  Nuremberg,  1468 ;  died  in  1528.  Wrote 

on  mathematics,  geography,  and  astronomy. 

Widmann,  Johann,  von  Eger.  Lived  in  1489.  Lectured  on  alge- 
bra at  Leipzig.  The  originator  of  German  algebra.  Wrote 
also  on  arithmetic  and  geometry. 


322  HISTORY  OF  MATHEMATICS. 

Witt,  Jan  de.  Born  in  1625,  died  in  1672.  Friend  and  helper  of 
Descartes. 

Wolf,  Johann  Christian  von.  Born  at  Breslau,  1679;  died  at 
Halle,  1754.  Professor  of  mathematics  and  physics  at  Halle, 
and  Marburg.  Text-book  writer. 

Woepcke,  Franz.  Born  at  Dessau,  May  6,  1826  ;  died  at  Paris, 
March  25,  1864.  Studied  the  history  of  the  development  of 
mathematical  sciences  among  the  Arabs. 

Wren,  Sir  Christopher.  Born  at  East  Knoyle,  1632  ;  died  at  Lon- 
don, in  1723.  Professor  of  astronomy  at  Gresham  College ; 
Savilian  professor  at  Oxford  ;  president  of  the  Royal  Society. 
Known,  however,  entirely  for  his  great  work  as  an  architect 


INDEX.' 


Abacists,  39,  41. 

Abacus,  15,  25,  26,  37. 

Abel,  62,  154,  155,  163,  181-188, 

Abscissa,  229. 

Abul  Wafa,  225,  286. 

Academies  founded,  116. 

Adelard  {^Ethelhard)  of  Bath,  74,  218. 

Africanus,  S.  Jul.,  202. 

Ahmes,  19,  31,  32,  34.  77,  78,  192,  282. 

Aicuin,  41. 

Al  Banna,  Ibn,  30,  76,  90. 

Al  Battani,  285. 

Alberti,  227. 

Algebra,  61,  77,  96,  107;   etymology, 

88 ;  first  German  work,  no. 
Algorism,  75. 

Al  Kalsadi,  30,  31,  75,  76,  89,  90,  92. 
Al  Karkhi,  75,  93- 
Al  Khojandi,  76. 
Al  Khowarazmi,  29,  33,  74,  75,  88.  89, 

91,  217. 
Al  Kuhi,  217. 
Alligation,  34. 
Almagest,  283. 
Al  Nasawi,  30,  34. 
Al  Sagani,  217. 
Amicable  numbers,  35. 
Anaxagoras,  195,  213. 
Angle,  trisection  of,  196,  197,  207,  208, 

217. 

Annuities,  56,  148. 
Anton,  I79». 
Apian,  108,  288,  289. 
Apices,  15,  27.  37,  39- 
Apollonius,  80,  152,  190,  200-209,  228, 

229,  231. 
Approximations  in  square  root,  70. 


Arabs,  3,  15,  20,  35,  39,  53,  74,  76,  88, 
89,  191,  214,  285. 

Arbitration  of  exchange,  55. 

Arcerianus,  Codex,  214,  218. 

Archimedes,  68-71,  78,  81-83,  190,  199, 
204,  205,  208,  2IO,  312. 

Archytas,  69,  82,  204,  207,  211. 

Argand,  124,  125. 

Aristophanes,  25. 

Aristotle,  64,  70. 

Arithmetic,  18,  24,  36,  49,  51,  64,  95. 

Arithmetic,  foundations  of,  189 ;  re- 
quired, 43. 

Arithmetical  triangle,  118. 

Aronhold,  146,  250. 

Aryabhatta,  12,  72,  74,  215,  216. 

Aryans,  12. 

Associative  law,  119. 

Assurance,  56-60. 

Astronomy,  18. 

August,  246. 

Ausdehnungylehre,  127. 

Austrian  subtraction,  28,  48. 

Avicenna,  76. 

Axioms,  197. 

Babylonians,  9,  10,   14,  19,  24,  25,  63, 

64,  190,  192,  193. 
Bachet,  106,  134,  137. 
Ball,  W.  W.  R.,  172*. 
Baltzer,  167*,  224». 
Bamberger  arithmetic,  51. 
Banna.     See  Ibn  al  Banna. 
Bardin,  277. 
Barrow,  169,  238. 
Bartl,  167. 
Barycentritcher  CaldU,  129,  250. 


'The  numbers  refer  to  pages,  the  small  italic  w's  to  footnotes. 


324 


HISTORY  OF  MATHEMATICS. 


Baumgart,  137*. 

Burgersckulen,  23. 

Beaune.    See  DeBeaune. 

Burgi,  4,  50,  98,  115,  n6,  290. 

Bede,  36,  37,  4°- 

Busche,  139. 

Bellavitis,  250,  266. 

Beltrami,  148,  269,  271. 

Calculating  machines,  48. 

Beman,  124*.,  i25».,  129*. 

Calculus,   differential,  168,    170,  171, 

Beman  and  Smith,  2O7». 

178;  directional,  127;  integral,  174. 

Benedictis,  225. 

178;   of  logic,   131;   of  variations. 

Bernecker,  109. 

179. 

Bernelinus,  37,  40. 

Cantor,  G.,  120,  123;  Cantor,  M.,  ~tn. 

Bernoulli  family,  58  ;  Jacob,  148,  150, 

Capelli,  165. 

152,  171,  175,  178,  179,  238,  239  ;  John, 

Cardan,  101-103,  109.  "2,  113,  150,  155, 

152,  166,  173,  175,  178,  179,  238,  242, 

225. 

243  ;  Daniel,  166,  175. 

Cardioid,  241. 

Bertrand,  122,  155,  270. 

Carnot,  174,  244,  246,  248. 

Bessel,  237. 

Cassini's  oval,  241. 

Betti,  165. 

Castelnuovo,  275. 

Beutel,  22. 

Cataldi,  131. 

B6zout,  143,  159,  160,  167. 

Catenary,  241. 

Bhaskara,  73,  74,  85,  86,  216. 

Cattle  problem  of  Archimedes,  83. 

Bianchi,  147. 

Cauchy,  62,  119,  124,  125,  138,  139,  143 

Bianco,  237*. 

153,  154.  i&t,   167,  168,   174,  181,  188. 

Bierens  de  Haan,  222«. 

189. 

Binder,  257. 

Caustics,  238. 

Binomial  coefficients,  103  ;  binomial 

Cavalieri,  168,  173,  224,  229,  234,  235 

theorem,  118. 

237. 

Biot,  242. 

Cayley,  i26».,  i2g».,  131,  143,  146,  168 

Boethius,  26,  27,  37,  215. 

178,  253,  257,  263,  264,  266,  274,  277. 

Bois-Reymond,  155,  189. 

Ceulen,  222. 

Boklen,  167*.,  270. 

Ceva,  244. 

Bolyai,  270,  271. 

Chain  rule,  52,  55. 

Bombelli,  101,  102,  112. 

Chance.    See  Probabilities. 

Boncompagni,  75. 

Chappie,  244. 

Bonnet,  155- 

Characteristics,  Chasles's  method  of, 

Boole,  131,  146. 

264. 

Bouniakowsky,  139. 

Chasles,  290*.,  246,  249,  256-258,  263- 

Bouvelles,  237. 

265. 

Boys,  166. 

Chessboard  problem,  135. 

Brachistochrone,  178,  238. 

Chinese,  8,  14,  19,  28,  74,  87,  214,  216. 

Brahmagupta,  52,  216. 

Christoffel,  147. 

Brianchon,  244. 

Chuquet,  47,  95. 

Briggs,  292. 

Church  schools,  3,  36,  37,  94. 

Brill,  I42».,  I75».,  i8o».,  189,  254,  264, 

Circle,  division  of,  24  ;  squaring,  195, 

276*.,  278. 

Bring,  165. 

Cissoid,  2H. 

Brioschi,  143,  144,  146. 

Cistern  problems,  34. 

Brocard,  245. 

Clairaut,  117,  242. 

Brouncker,  134. 

Clausberg,  55. 

Brune,  59. 

Clavius,  in. 

Brunelleschi,  227. 

Clebsch,  146,  147,  176,  177,  250,  25iw., 

Burckhardt,  134,  141,  147. 

257,  262,  266,  279. 

325 


Cloister  schools.  See  Church  schools. 

Codex  Arcerianus,  214. 

Coefficients  and  roots,  115,  156. 

Cohen,  172*. 

Cole,  i62«. 

Combinations,  70,  74, 150, 151. 

Commercial  arithmetic,  22,  51,  60. 

Commutative  law,  119. 

Compasses,  single  opening,  225. 

Complementary  division,  38. 

Complex  numbers,  73,  101,  123,  126, 
182.  Complex  variable.  See  Func- 
tions, theory  of. 

Complexes,  254. 

Compound  interest,  52. 

Computus,  37,  39. 

Conchoid,  211. 

Condorcet,  149. 

Conies,  81,  202,  204-208,  228,  230,  239, 
256. 

Congruences,  theory  of,  131. 

Conon,  210. 

Conrad,  H.,  109. 

Conrad  of  Megenberg,  219. 

Contact  transformations,  178, 269,  276. 

Continued  fractions,  131-133,  168. 

Convergency,  152-155,  189.  See  Se- 
ries. 

Coordinates,  Cartesian,  231;  curvi- 
linear, 268,  269;  elliptic,  269. 

Copernicus,  289. 

Correspondence,  one-to-one,  251,  264, 


Cosine,  288. 

Coss,  96-99,  107,  109,  in. 

Cotes,  174,  239,  a*'.  244- 

Coifnting,  6. 

Cousin,  227. 

Covariants,  146.  See  also  Forms,  In- 
variants. 

Cramer,  132.  167,  240;  paradox,  240. 

Crelle,  141,  245,  257. 

Cremona,  256,  266. 

Crofton,  276. 

Cross  ratio,  258,  259. 

Cube,  duplication  of,  82,  104,  204,  207; 
multiplication  of,  207,  211. 

Culvasutras,  72. 

Cuneiform  inscriptions,  9. 

Cunynghame,  166. 

Curtze,  28g». 


Curvature,  measure  of,  268. 

Curves,  classification  of,  233,  239,  246; 
deficiency  0^262,263;  gauche  (of 
double  curvature),  243,  255,  263; 
with  higher  singularities,  253. 

Cusa,  237. 

Cycloid,  178,  237,  238. 

d,  symbol  of  differentiation,  170-172 ; 

8,  symbol  of  differentiation,  180. 

D'Alembert,  175,  180. 

Dante,  94. 

DeBeaune,  156. 

Decimal  fractions,  50. 

Decker,  292. 

Dedekind,  120-122,  126,  127,  189. 

Defective  numbers,  35. 

Deficiency  of  curves,  262, 263. 

Definite  integrals,  174. 

Degrees  (circle),  24. 

De  Lagny,  157. 

De  la  Gournerie,  261. 

Delambre,  295. 

De  1'Hospital,  173,  178,  179. 

Delian  problem,  82,  104,  204,  207. 

Democritus,  213. 

De  Moivre,  124, 152,  160. 

De  Morgan,  143,  155. 

Desargues,  205,  237,  242,  259. 

Descartes,  4,  108,  117,  119,  124,  136, 

140,  156, 191,  228,  230-233,  238. 
Descriptive  geometry,  247,  259,  260. 
Determinants,  133,  144,  145,  167,  168, 

262. 

DeWitt,  57,  148. 
Dialytic  method,  144,  145. 
Diametral  numbers,  105. 
Differential  calculus, 168,  170, 171,178; 

equations,  174-178,  269 ;  geometry, 

867. 

Dimensions,  «.,  275. 
Dini,  155,  189. 
Dinostratus,  1971  210. 
Diocles,  211. 
Diophantus,  65,  70,  77.  81,  84,  85,  90, 

93-  133,  134- 
Dirichlet,  62,  125,  126,  133,  139,  140, 

153,  174.  177,  181,  189,  279. 
Discount,  54. 
Discriminant,  145. 
Distributive  law,  119,  130. 


326 


HISTORY   OF  MATHEMATICS. 


Divani  numerals,  15. 

Eudoxus,  79,  199,  204,  210,  212,  223. 

Divisibility  tests.  35. 

Euler,  58,  62,  118,   124,   132.  135    136' 

Division,  38,  42,  44,  48,  49. 

138,  140,  143,  152-154,  158,  160,  173, 

Dodson,  58. 

175,  179,  180-182,  240,  244,  247,  167, 

Donatello,  227. 

294,  295. 

Duality,  249. 

Evolutes,  238,  242. 

DuBois-Reymond,  155,  189. 

Exchange,  52,  55. 

Duhamel,  155. 

Exhaustions,  199,  225. 

Duodecimal  fractions,  19. 

Exponents.     See  Symbols. 

Dupin,  267,  270. 

Eyck,  226. 

Duplication  of  the  cube,  82,  104,  204, 

Eycke,  222. 

207. 

Durer,  221,  224-227. 
Dyck,  278. 

Fagnano,  180,  i8t. 

Farr,  59. 

f,  irrationality  of,  133. 

Faulhaber,  96. 

Easter,  41. 

Felkel,  141. 

Ecole  polytechnique,  261. 

Fermat,  57,  118,  134,  135,  137,  140,  -  ;' 

Eccentricity,  224. 

168,  173,  229,  234. 

Egyptians,  8,  10,  18,  24,  31,  35,  63,  77, 

Ferrari,  112,  155,  225. 

190,  192,  282. 

Ferro,  112. 

Eisenstein,  126,  127,  138. 

Feuerbach,  245. 

Elimination,  theory  of,  142,  143. 

Fibonacci.    See  Leonardo. 

Ellipse,  81,  205. 

Finck,  288. 

Ellipsoid,  242. 

Finger  reckoning,  25,  36,  43. 

Elliptic  functions.    See  Functions. 

Fischer,  59. 

Elliptic  integrals,  classed,  183,  186, 
187. 

Fluxions,  171,  173. 
Forms,  theory  of,  131,  142-147. 

Ellis,  131. 

Fourier,  153. 

Enneper,  i8iw. 
Enumerative  geometry,  264. 

Fourth  dimension,  274. 
Fractional  exponents,  102. 

Envelopes,  242. 

Fractions,  31,40,  49;  continued,  131- 

Equations,   approximate   roots,  156, 

133,  168;  sexagesimal,  282-284;  duo- 

166; Abelian,  163;  cubic,  81,  82,  92,  93, 

decimal,  19. 

111-113,   155;    cyclotomic,    160-163, 

Francais,  125. 

207;    differential,   174-178;    funda- 

Frenicle, 106. 

mental  theorem,   163;    higher,  92, 
"5.  i55-i6°r  164-166;  indeterminate, 

Fr6zier,  260. 
Frobenius,  177,  178,  189. 

83,  84,  86,  93.  135.  137,  139  ;  linear, 
77,  78,  87,  90  ;  limits  of  roots,  156, 
166;  Diophantine,  93,  135,  137;  quad- 
ratic, 79-81,85,  91,  109,  155;  quartic, 
111-113;  quintic,   165;   mechanical 

Fuchs,  177,  178,  181. 
Functional  determinants,  168. 
Functions,  Abelian,  180,  186,  188,  189; 
elliptic,  165,  180-182  ;  periodicity  of, 
184;  symmetric,  142,  143;  theory  of, 

solutions,  166;  modular,  164  ;  nega- 
tive roots,  234. 

177,  180,  181,  188;  theta,  182,  188,  189 
Fundamental  laws  of  number,  IIQ, 

Equipolent,  96. 

131,  189. 

Eratosthenes,  141,  190,  208. 

Erchinger,  162. 

Galileo,  237,  241. 

Eschenbach,  151. 

Galois,  164. 

Euclid,  35,  65-^9,  79,  80,  100,  119,  133, 

Gauss,  4,  124-128,  133,  I35».,   136-140, 

190,  iQ5,  197-199,  212,  213. 

142,  143,  145,  149,  15°,  153,  154,  156' 

327 


160-163,  167,  174,  181,  188,  207,   245, 
267,  270,  275,  279,  294,  295. 

Geber,  286. 

Gellibrand,  292. 

Geminus,  an. 

Genocchi,  139. 

Geometric  means,  78,  103 ;  models, 
276. 

Geometry,  66,  190,  314;  analytic,  igi, 
205,  230,  232,  246;  descriptive,  247, 
259,  260;  differential,  267;  enumera- 
tive,  264  ;  metrical,  190,  192,  193 ; 
protective,  191,  246,  247,  258;  non- 
Euclidean,  270;  of  position,  190, 246, 
248,  258;  of  space,  211,  242;  three 
classes  of,  274. 

Gerbert,  15,  37,  40,  61,  218. 

Gergonne,  249,  257. 

Gerhard  of  Cremona,  40,  286. 

Gerhardt,  47«. 

German  algebra,  96,  107 ;  universi- 
ties, 95. 

Giesing,  io6». 

Girard,  124. 

Girls'  schools,  21. 

Gizeh,  9. 

Glaisher,  142. 

Gmunden,  95. 

Gnomon,  66,  92,  195,  213. 

Goepel,  188. 

Golden  rule,  51. 

Golden  section,  195,  222,  223. 

Gordan,  144,  146,  147. 

Gournerie,  261. 

Goursat,  178. 

Gow,  7«. 

Grammateus,  45,  49,  98,  99,  108,  109. 

Grassmann,  127-129,  131,  256,  275. 

Graunt,  57. 

Grebe  point,  245. 

Greek  fractions,  32. 

Greeks,  2,  8,  10,  14,  19,  20,  25,  64,  77, 
190,  193,  282. 

Gregory,  151. 

Groups,  theory  of,  164,  177;  point,  240 

Grube,  23. 

Grunert,  '28,  257. 

Gubar  numerals,  15,  17,  31. 

Gudermann,  183. 

Guilds,  56. 

Guldin,  213,  224,  334 


Gunter,  288. 
Gvinther,  i6 


.,  133,  168,  93ion 


Haan,  222*. 

Hachette,  361. 

Hahn,  48. 

Halley,  57,  58,  166,  203,  204. 

Halphen,  147,  253,  256,  264,  269. 

Hamilton,  137,  270. 

Hammer,  295*. 

Hankel,  6«.,  124,  247/2. 

Harmonic  means,  78,  79. 

Harpedonaptae,  193,  194. 

Harriot,  101,  117,  156. 

Hebrews,  10. 

Heine,  120,  133,  133,  189. 

Helix,  211,  343. 

Helmholtz,  271,272. 

Henrici,  277. 

Heptagon,  336. 

Hermite,  133,  146,  147,  165. 

Herodotus,  34. 

Herodianus,  n. 

Heron,  64,  70,  78,  81,  84,  301,  312,  283. 

Hess,  345. 

Hesse,  143-145.  164.  168,  176,  344,  250, 

262. 

Hessel,  245. 

Heteromecic  numbers,  67. 
Hexagram,  mystic,  337,  244. 
Heyn,  59. 

Hieratic  symbols,  g. 
Hilbert,  147,  148. 
Hindenburg,  132,  150. 
Hindu  algebra,  84;  arithmetic,  34,71. 

72;    fractions,  33;    geometry,  214; 

mathematics,  2,  12  following. 
Hipparchus,  213,  366.  382,  283. 
Hippias,  196,  310. 
Hippocrates,  65,  83,  197,  204,  313. 
HSlder,  189. 
Homology,  349. 
Hoppe,  167,  I73»-,  345- 
Hospital,  173,  178,  179. 
Homer,  166. 
Hudde,  108,  148,  156. 
Hugel,  107. 
Hurwitz,  364. 

Huygens,  131,  148,  222,  238,  343. 
Hyperbola,  81,  205. 
Hyperboloid,  242. 


328 


HISTORY  OF  MATHEMATICS. 


Hyperdeterminants,  146. 
Hyperelliptic  integrals,  187. 
Hypergeometric  series,  153. 
Hypsicles,  84,  200,  212. 

i  for  1^,  124. 

lamblichus,  136. 

Ibn  al  Banna,  30,  76,  90. 

Ibn  Kurra,  136,  217. 

Icosahedron  theory,  166. 

Ideal  numbers,  126. 

Imaginaries.   See  Complex  numbers. 

Incommensurable  quantities,  69. 

Indeterminate  equations.  See  Equa- 
tions. 

Indivisibles,  234,  236. 

Infinite,  173.    See  Series. 

Infinitesimals,  169,  170,  173, 174. 

Insertions,  208,  211. 

Insurance,  56-58. 

Integral  calculus,  174,  178. 

Interest,  54. 

Invariants,  145-148,  262,  274. 

Involutes,  238,  241. 

Involutions,  252. 

Irrational  numbers,  68,  69,  100,  119, 
122,  123,  133,  189. 

Irreducible  case  of  cubics,  112. 

Isidorus,  36. 

Isoperimetric  problems,  179,  200. 

Italian  algebra,  90. 

Jacobi,  62,  138,  139,  143,  144,  165,  168, 
174-177,  181-187,  269-  276,  279. 

Johann  von  Gmunden,  95. 

Jonquieres,  256. 

Jordan,  165. 

Kalsadi.    See  Al  Kalsadi. 

Karup,  56,  59. 

Kastner,  48. 

Kepler,  4,  50,  61,  169,  173,  191,  222-224, 

245,  288. 

Khayyam,  75,  89,  92,  93. 
Khojandi,  76. 

Khowarazmi.  See  Al  Khowarazmi. 
Klein,  147,  165,  177,  178,  2O7».,  254,  274, 

277,  278. 
Knilling,  23. 
KBnigsberger,  180. 
Kossak,  I2ow. 


Krafft,  135. 

Kronecker,  139,  165. 

Kruger,  141. 

Krumbiegel  and  Amthor,  83. 

Kummer,  126,  138,  i39».,  155,  270,  278. 

Kurra,  Tabit  ibn,  136,  217. 

Lacroix,  242,  261. 

Lagny,  De,  157. 

Lagrange,  62,  136,  138,  143,  151,  i59 
160,  166,  167,  173,  175,  176,  179,  j8o. 
182,  239,  267,  294,  295. 

Laguerre,  274. 

Lahire,  106,  249. 

Lalanne,  167. 

Laloubfcre,  158. 

Lambert,  124,  133,  141,  260,  267,  295 

Lam6,  240,  269. 

Landen,  180,  182,  244. 

Lansberg,  249. 

Laplace,  150, 151,  167,  175. 

Latin  schools,  21,  43. 

Least  squares,  149. 

Lebesgue,  139. 

Legendre,  133,  136,  138-140,  149,  166, 
174,  180-184,  187,  270,  295. 

Lehmus,  257. 

Leibnitz,  4,  48,  54,  58,  62,  117,  150-15- 
156,  167,  170-173,  178,  229,  239,  242. 

Lemniscate,  241. 

Lencker,  227. 

Leonardo  da  Vinci,  225  ;  of  Pisa  (Fi- 
bonacci), 40,  41,  45,  95,  101,  107,  ice,-, 
in,  218. 
Leseur,  158. 
Lessing,  83. 

Letters  used  for  quantities,  64. 
Lexell,  295. 

L' Hospital,  173,  178,  179. 
Lhuilier,  244. 
Lie,  147,  177,  269,  276. 
Lieber,  245». 
Light,  theory  of,  270. 
Limacon,  241. 

Limits  of  roots,  156,  160,  166. 
Lindemann,  133,  189,  207. 
Liouville,  139,  181,  269. 
Lipschitz,  147. 
Lituus,  241. 
Lobachevsky,  271. 
Loci,  209,  210,  232. 


INDEX. 


329 


Logarithmic  series,  151;  curve,  241. 

Morgan,  59. 

Logarithms,  290. 

Mortality  tables,  57,  148. 

Logic,  calculus  of,  131. 

Moschopulus,  106. 

Logistic,  64. 

Muir,  167*. 

Loria,  240*. 

Miiller,  47*. 

f.oxodrome,  243. 

Multiplication,  45,  46. 

Luca  Pacioli.    See  Pacioli. 

Muret,  277. 

Lunes  of  Hippocrates,  197. 

Mystic  hexagram,  237,  244. 

Liiroth,  i68». 

Mysticism.    See  Numbers. 

Maclaurin,  152,  156,  174,  180,  238,  239- 

Nachreiner,  168. 

Macrobius,  36. 
Magic  squares,  54,  105-107. 

Napier,  47,  172,  288,  290. 
Nasawi,  30,  34. 

Magnus,  265,  277. 

Negative  numbers  and  roots,  70,  72, 

Majer,  2io». 

So,  91,  101,  109,  119. 

Malfatti,  159,  256. 

Neo-Platonists,    68;    -Pythagorean?, 

Malus,  270. 

68. 

Marie,  230^. 

Netto,  i62». 

Marre,  gdn. 

Neumann,  C.,  269;  K.,  57. 

Mathematica,  64. 

Newton,  4,  62,  H7-«9>  152,  156,  166, 

Matthiessen,  77«.,87».,  io8». 
Maxima,  169,  179,  180,  203. 

170-175,  178,  234,  239. 
New  Zealanders,  7. 

Mean-value  theorems,  189. 

Nicomachus,  78. 

Means,  geometric  and  harmonic,  78, 

Nicomedes,  210. 

79- 

Nines,  casting  out,  35,  46,  76. 

Mehmke,  167, 

Noether,  I44»-,  165,   180*.,   189,   253, 

Meister,  244. 

256,  264,  266. 

Menaechmus,  82,  204-207. 

Non-Euclidean  geometry,  270. 

Menelaus,  283. 

Normal  schools,  23. 

Menher,  HI. 

Numbers,  amicable,  136;  classes  of, 

Mercator,  151. 

67;  concept  of,  118,  120;  ideal,  126; 

Merchants'  rule,  51. 

irrational,  68,  69,  100,  119.  122,  123, 

Merriman,  149)*. 

133,  189;  mysticism  of,  37,  106;  na- 

Method, 23. 

ture  of,  118,  120;  negative,  70,  101, 

Meusnier,  243,  267. 
Meyer,  F.,  275;  W.  F.,  147;  -Hirsch, 

109;  perfect,  35,  68;  polygonal,  71  ; 
prime,  67,  68,  136,  141,  161,  162;  py- 

143- 
Meziriac,  106,  134,  *37- 

ramidal,  71;  plane  and  solid,  66; 
systems  of,  6  ;  theory  of,  133-14°- 

Middle  Ages,  3,  20,  44,  51,  56,  i°6,  *5i- 

Numerals,  6. 

Minima,  169,  179,  180,  203. 

Nunez,  in,  243. 

Minus.    See  Symbols. 

Nuremberg,  21. 

Mobius,  128,  129,  133,  244,  249,  250-252, 

258,  263,  265,  295. 

Models,  geometric,  276. 

Oddo,  39. 

Mohammedans,  3.    See  Arabs. 

Oekinghaus,  167. 

Moivre,  124,  152,  160. 

Oenopides,  195. 

Mollweide,  106. 

Olivier,  261. 

Mommsen,  11. 

Omar  Khayyam,  75,  89,  92,  93. 

Monge,  176,  178,  247,  248,  267,  277. 

One-to-one  correspondence,  251,  264, 

Monks.    See  Church  schools. 

266,  268. 

Montucla,  69*. 

Ordinate,  229. 

330 


HISTORY  OF  MATHEMATICS. 


Oresme,  95,  102,  829, 

Osculations,  239. 
Oughtred,  117, 156. 


IT,  nature  of,  133,  207;  values  of,  192, 

'93,  199.  201,  2I5-2I8,  222. 

Pacioli,  42,  45-47.  52,  95,  96,  101. 

Page  numbers,  16. 

Pappus,  65,  179,  202,  203,  208,  209,  212, 

234. 

Parabola,  81;  area,  68;  name,  205. 
Paraboloid,  242. 
Parallel  postulate,  201,  270. 
Parameter,  205. 
Parent,  242,  247. 
Partition  of  perigon,  160-162. 
Partnership,  34. 
Pascal,  48,  57,  118,  148,  150,  169,  173, 

174-  234,  236-238. 
Pascal's  triangle,  118,  150. 
Pauker,  155,  161. 
Peirce,  131. 
Peletier,  in. 
Pencils,  242. 
Pepin,  139. 

Perfect  numbers,  35.  68. 
Periodicity  of  functions,  184. 
Permutations,  74. 
Perspective,  226,  227,  259. 
Pessl,  107. 
Pestalozzi,  23. 
Petersen,  139. 
Petty,  57- 

Peuerbach,  3,  42,  45,  103,  289. 
Pfaff,  151,  153.  175,  176- 
Philolaus,  78. 
Phoenicians,  8, 10. 
Piazzi,  149. 
Pincherle,  189. 
Pitiscus,  so».,  290. 
Pitot,  243. 
Plane  numbers,  66. 
Plato,  67,  82,  197,  207 ;  of  Tivoli,  285. 
Platonic  bodies,  212. 
Pliny,  26. 
Plucker,  144,  239,  249-252,  254,  256, 257, 

265,  275,  277. 

Pliicker's  equations,  253. 
Plus.    See  Symbols. 
PoStius,  141. 


Poincare1,  165,  177. 

Poinsot,  245. 

Point  groups,  240. 

Poisson,  143, 173. 

Polar,  249,  256. 

Pole,  249. 

Political  arithmetic,  56. 

Polygons,  star,  218,  219,  224. 

Polytechnic  schools,  261. 

Poncelet,  246,  248,  249,  252,  258,  265. 

Position  arithmetic,  17. 

Pothenot,  295. 

Power  series,  103. 

Powers  of  binomial,  118. 

Prime  numbers,  67,  68,  136,  141,  161, 

162. 

Pringsheim,  I54«.,  155,  189. 
Prismatoid,  246. 
Probabilities,  148,  149,  276. 
Proclus,  219. 

Projection,  213,  214.    See  Geometry. 
Proportion,  79,  109. 
Ptolemy,  201,  214,  266,  283. 
Puzzles,  54. 

Pythagoras,  68,  179,  190,  194,  195,  214 
Pythagoreans,  35,  66,  67,  78,  136,  194, 

195,  198. 

Quadratic  equations.  See  Equations. 
Quadratic  reciprocity,  137,  138;    re- 
mainders, 76. 
Quadratrix,  196,  241. 
Quadrature  of  circle.    See  Circle. 
Quadrivium,  94. 
Quaternions,  127,  129. 
Quetelet,  59. 

Raabe,  155. 

Radicals,  100. 

Rahn,  96». 

Ramus,  98,  in,  133. 

Raphson,  166. 

Realschulen,  23. 

Reciprocity,  quadratic,  137, 138;   Her 

mite's  law  of,  146. 
Reciprocal  polars,  249. 
Reckoning  schools,  4. 
Redundant  numbers,  35. 
Rees,  55. 
Regeldetri,  34,  51. 


Regiomontanus,  3,  42,  107,  108,  219, 

Saurin,  244. 

287,  289,  294- 

Scalar,  130. 

Regulae,  various,  34,  41,  51,  52,  54,  90, 

Scheeffer,  189. 

"5- 

Scheffler,  59,  127,  130,  245,  257. 

Regular  polygons,  161,  162,  221,  223, 

Schellbach,  257. 

225,  226,  237,  245  ;  solids,  212. 

Schering,  139. 

Reiflf,  I5i».,  178*. 

Scheubel,  98,  in. 

Reinaud,  75. 

Schlegel,  i27».,  345. 

Resolvents,  159. 

Schlesinger,  174  «. 

Resultant,  i43-*45- 

Schooten,  Van,  136,  141,  156,  242. 

Reuschle,  142,  167. 

Schottky,  189. 

Reymers,  96,  98,  107,  108,  115. 

Schroder,  131. 

Rhabda,  25. 

Schubert,  246,  264,  275. 

Rhaeticus,  288. 

Schwarz,  178,  278. 

Riccati,  175. 

Schwenter,  131,  226. 

Riemann,  62,  153,  154*.,  181,  188,  189, 

Scipione  del  Ferro,  112. 

271,  272,  275,  276. 

ScOtt,  240». 

Riese,  97,  99,  106,  no,  113,  114,  120. 

Secant,  288. 

Right  angle,  construction  of,  219. 

Seelhoff,  i36».,  140*. 

Roberval,  169,  173,  229,  234,  236,  238. 

Segre,  275. 

Rodenberg,  278. 

Seidel,  154. 

Rohn,  278. 

Semitic,  9. 

Rolle,  158. 

Seqt,  282. 

36,  37;  mathematics,  2,  8,  19,  214. 

Series,  34,  67,  71,  74,  76,  103,  151  154, 
189. 

73,  103;  negative,  234;  real  and  im- 

Servois, 249. 

aginary,    124,  see    also    Numbers, 

Sexagesimal  system,  24,  25,  34,  64,  70, 

complex;    square,   69,   70,   73,   103. 

282-284. 

See  also  Equations. 

Sieve  of  Eratosthenes,  67. 

Rope  stretchers,  193  ;  stretching,  215. 

Signs.    See  Symbols. 

Roriczer,  220. 

Simpson,  166. 

Rosanes,  266. 

Sine,  name,  285. 

Rosenhain,  188. 

Skew  determinants,  168. 

Rosier,  i2ow. 

Smith,  D.  E.,  I78».    See  Beman  and 

Roth,  96,  106. 

Smith.    H.  J.  S.,  253. 

Rothe,  132,  151. 

Snellius,  222,  243,  295. 

Rudel,  245. 

Soleil,  277. 

Rudio,  222«. 

Solid  numbers,  66. 

Rudolff,  4,  50,  53,  97-100,  109-111,  113- 

Sonnenburg,  223». 

115. 

Spain,  3. 

Ruffini,  163. 

Spirals,  241  ;  of  Archimedes,  210. 

Rule  of  three,  34,  51.   See  Regulae. 

Squares,  least,  149. 

Squaring  circle.    See  Circle. 

f,  symbol  of  integration,  170,  172. 
Saint-Vincent,  151. 

Stahl,  189. 
Star  polygons,  218,  219,  224. 

Salignac,  in. 

Steiner,  225,  246,  249,  251,  256-258,  265. 

Salmon,  143,  263. 

Stereographic  projection,  266. 

Sand-reckoner,  71. 

Stereometry,  211,  224. 

Sanskrit,  12,  13. 

Stern,  133,  139- 

Sauce,  269. 

Stevin,  50,  228. 

332 


HISTORY  OF  MATHEMATICS. 


Stewart,  244. 

Thompson,  107,  266. 

Stifel,  4,  49,  52,  53,  97,  99-105,  109-111, 

Timaeus,  2ia. 

113,  115,  Il8,  220,  221,  224. 

Tonti,  56. 

Stokes,  154. 

Tontines,  57. 

Stoll,  246. 

Torricelli,  237. 

Stolz,  i2o». 

Torus,  213. 

Stringham,  245. 

Transformations  of  contact,  178,  269 

Stubbs,  266. 

276. 

Sturm,  48,  270. 

Transon,  270. 

Substitutions,  groups,  164,  165. 

Transversals,  144,  248. 

Sun  tse,  87. 

Trenchant,  47. 

Surfaces,  families  of,  267;  models  of, 

Treutlein,  52*1.  ,  67,  96*.,  97*. 

277;    of    negative    curvature,   273; 

Trigonometry,  281. 

second  order,  213,  262  ;  third  order, 

Trisection.     See  Angle. 

263  ;  skew,  255  ;  Steiner,  256;  ruled, 

Trivium,  94. 

255- 

Tschirnhausen,  157,  159,  165,  178,  238, 

Surveying,  18,  71. 

241,  242. 

Suter,  94*. 

Tylor,  6m. 

Swan  pan,  28. 

Sylow,  165. 

Ubaldi,  228. 

Sylvester,  143-147,  276- 

Ulpian,  56. 

Sylvester  II.,  Pope,  15. 

Unger,  i6w. 

Symbols,  47,  63,  65,  71.  76,  88,  89,  95- 

Universities,  rise  of,  94. 

97,  99,  102,  108,  109,  117,  170,  171,  183, 

Unverzagt,  i2g».,  isow. 

'97- 

Symmedians,  245. 

Symmetric  determinants,  168;  func- 

Valentiner, 256. 

tions,  142,  143. 

Van  Ceulen,  222. 

Van  der  Eycke,  222. 

Tabit  ibn  Kurra,  136,  817. 

Vandermonde,  118,  159,  167. 

Tables,  astronomical,  286;    chords, 

Van  Eyck,  226. 

282;    factor,    141;    mortality,    148: 

Van  Schooten,  136,  141,  156,  242. 

Variations.    See  Calculus. 

143;  sines,  286;  theory  of  numbers. 

Vector,  130. 

142;   trigonometric,  282,  286,    289, 

Vedas,  25. 

290,  293. 

Veronese,  275. 

Tacquet,  174. 

Versed  sine,  288. 

Tanck,  23. 

Victorius,  27. 

Tangent,  288. 

Vieta,  107,  108,  115,  117,  119,  134,  156, 

Tannery,  33,  70,  120. 

191,  222,  229,  249,  287,  288. 

Tartaglia,  3,  49,  51,  52,  112,  115,  155, 

Vincent,  St.,  151. 

225. 

Vitruvius,  215. 

Tatstha,  29. 

Vlacq,  292. 

Taylor,  B.,  152,  166,  259;  C.,  224*. 

Voigt,  139. 

Thales,  194. 

Von  Staudt,  162,  246,  249,  257-259,  263 

Theaetetus,  212. 

Vooght,  292. 

Theodorus,  69. 

Theon  of  Alexandria,  34,  70. 

Wafa,  225,  286. 

Thieme,  244. 

Wallis,  117,  125,  131,  135,  154,  173,  23  ( 

Thirty  years'  war,  22. 

236.  237,  242. 

Thome1,  177. 

Waring,  143,  159,  239. 

INDEX. 


333 


Weber,  189. 

Weierstrass,  62,  120,  147,  178,  181,  189. 

Welsh  counting,  8 ;  practice,  53. 

Wessel,  125. 

Widmann,  47,  51,  220. 

Wiener,  2z6«.,  245,  278. 

Witt,  De,  57,  148. 

Wittstein,  59,  256*1. 

Wolf,  47,  48. 

Woodhouse,  178*. 

Woolhouse,  276*. 


Wordsworth,  12*. 
Wren,  243,  247. 

x,  the  symbol,  97. 
Year,  length  of,  24. 

Zangemeister,  n. 
Zeller,  139- 
Zenodorus,  200. 
Zero,  12,  16,  39,  40,  74. 
Zeuthen,  68».,  253,  264- 


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